[N.B. This file is a transcript of the talk editied for reading clairity. A recording and transcript are available if needed. Additional minor editing done on 12/31/2014.]
Quantum mechanics as geometry: Conflict, Mechanism, Interpretation, and Implication
Good evening, I am happy to be here in Sweden.
As a theorist, I study geometrical theories of quantum mechanics. This effort started in the late sixties and has continued to the present. From the beginning, the goal has been to learn how to connect to other geometrical theories of physics, particularly gravity. Today, I would like to look at the special difficulties that arise when trying to do this. Combining the very mechanistic character of geometry with the essential statistics of quantum theory is the major conflict. To explain to you how this may be resolved is the principle goal of this talk.
Quantum mechanics and Gravity
After this short introduction, there are three parts to the presentation. I give, first, an elementary discussion of the geometrical approach as it has been developed. An essential ingredient is the electromagnetic field, which links gravitational and quantum theories. There is then a consideration of how statistics can come about in a mechanistic geometry. Finally, I will look at some of the implications of this development. Since this meeting is dedicated to quantum foundations, I will emphasize those parts of the theory that are relevant. Many of the paradigms of fundamental quantum mechanics are revisited.
Common aspects of gravity and quantum mechanics
While the sub-fields of physics are different, certain mathematical properties remain common. The field theoretic approach is essential and the use of second order differential equations pervasive. Gravity, electrodynamics, and quantum theory possess this commonality. It has been found possible to use Riemannian systems to combine the subfields. The equations are obtained from versions of the Riemann tensor in a suitable number of dimensions. Physical understanding has arrived at the point where certain questions must be asked. Hard conflicts have not been found but many new experiments are needed.
Physical development of multiple dimensions.
Spaces of higher dimension allow the introduction of additional fields.
In the simplest case of general relativity, remnants of quantum theory appear in the conformal parameter. A further extension by Weyl, brings in electrodynamics, but in a limited way.
A better approach is to use five dimensions to arrive at a theory that has both motion and field equations. The particles are charged, fixed rest mass, spin zero and move according to the Klein-Gordon equation including gravitational and electromagnetic forces. In this case, four of the five coordinates can be chosen to coincide with the four coordinates of space-time.
The eight dimensional theory presupposes a coordinate system with spinor transformation properties. The eight coordinates combined into complex pairs must be mapped to five space with the help of the Dirac matrices. In this case, none of new the spinor coordinates can be taken coincident with any of the space-time coordinates. Charged spin-1/2 electrons appear with neutrinos and the corresponding weak interactions. The possibilities are still under study; additional interactions are present in a complicated geometry. Other parts of the phenomenology of particle physics are expected to be part of this size geometry.
Wave-functions
The wave properties are found in the characteristic equations of the conformal parameter. All interactions are mediated by changes in the scale of the space as it has been assigned to each particle. In this sense, the wave-function is a representation of an entire space squeezed into a more or less compact mathematical object. No additional representation is needed.
Quantum essentials
These are sometimes called wave particles. They have a regular evolution as fields in the associated coordinate system. There are no classical point objects. Any Hilbert space is subsidiary to the differential field equation. A wave function may be expanded but only the sum total is attached to a structure in the geometrical system.
Mechanics, causality, and determinism
At this point, there are no statistics nor any measurement theory. Any macroscopic time direction comes from thermodynamics. The evolution is mechanistic and time symmetric. The implicit concepts of causality and determinism depend on the effect of specific decisions or events. Notice that the electromagnetic interaction is different. It travels at the speed of light, that is on null lines. Electromagnetism is used to define the null lines of general relativity. It has also always been essential to measurements in quantum theory.
Simple radiation
To see the problem, let us start with a simple example of the process of radiation in an inertial frame. A point mass on a spring oscillates with frequency omega. Radiation is emitted, with total power given by the standard formula. For quantum emission, we say that the vacuum interacts with the charge, connecting it to the radiation field.
Intermediate radiation
If this same particle were hung from a string in a gravitational field we would be assured that there would be no radiation. Yet this is not in an inertial frame, it has upward acceleration of magnitude g. If the particle is allowed to fall, it is now in an inertial frame but would be considered to radiate. The theoretical discussion of an accelerated article is ongoing, especially in the classical case. Of course if the particle is put on a fixed height tower on a rotating earth, the radiation will be from the rotation and not the gravity. Raising the tower increases the radiation rate. At the geosynchronous point, the motion becomes inertial. If the tower is removed, to lowest order the radiation is unchanged, while the energy comes now from gravitational potential energy rather than rotational kinetic energy. The simple concept of radiation cannot be adapted to these complexities.
General radiation
In the most general case, a charged mass point moves inertially among gravitating bodies. There is no characteristic inertial frame, no defined acceleration, no characteristic distance scale, no way to assure a velocity limit, no guarantee of simple E1 radiation, and no characteristic frequency or wavelength. A proper theory must handle these cases as well, including the possibility of other non-inertial forces.
Two point tensors
To resolve this enigmatic situation, the approach of gravitational theory is to use the two point tensor. It is a mathematical object, a type of Green's function, that depends on two points in space-time. It has tensor indicies which transform covariantly at either end. The radiative interaction can thus be described in a systematic way. Here a current at point x is coupled to a current at point x'. At a particular point, x, the sum of effects propagated from all points x' become the effective vacuum at x.
Radiation reaction
The calculation of radiation reaction has always been problematic. It is often calculated after the radiation is known. Here we have an example based on two closely spaced antennas. Each antenna alone has a power that depends on the average of the its driven current squared. If both antennas are turned on at the same time, the power is increased to correspond to the average of the square of the sum of the currents. The increased power must be supplied electronically by the transmitters driving the antennas. The simple calculation shows that the the added emission is equivalent to the energy required to move the current of each antenna through the radiative reaction field of the other. The forces of radiative reaction must be accepted as ontological. This effect of radiative reaction on nearby particles has been know for atomic spectroscopy from the early days.
Two point tensors (reprise)
If we go back to the previous slide, it is easy to argue that the radiative reaction forces must be symmetrical between particles and must be included explicitly in the geometry. All forces come from a distortion of the space-time and cannot be added in a later step in the calculation. They must be intrinsic to the calculation from the outset.
Normal radiative behavior
Even though a closed form quantum electrodynamics requires the gravitational formalism, the quantum electrodynamic expansion provides a helpful point of view. We begin with a perfectly reflecting empty cavity. The evolution of the field modes inside is mechanistically deterministic. Particles are added and the quantum equations will complete the system. The system is deterministic and free of statistics. As the number of particles increases, k of the particles become the experiment itself and n of the particles become the absorber. These remain around the edges and exchange photons with the experiment. As n becomes very large, normal radiative behavior is obtained. The character of the vacuum field does not depend on the mirror which may be removed. The system remains deterministic even though the experiment emits photons spontaneously. The randomness of spontaneous emission comes from the multiple interactions of the absorber. In this way, an explicit absorber, n>>1, can account for the statistical properties of the quantum emission process. The randomness occurs even though the complete system is mechanistically determined. The statistics appear in a way that is compatible with a wholly mechanistic geometry.
Quantum emission
The advanced interactions provide the initiation of the electromagnetic decay process. The uncertainty in the effect of the advanced field appears as the randomness of the spontaneous emission.
Born's experiment
We must now attempt to understand the statistics associated with a probability wave. We review Born's experiment. A quantum electron emerges from a nucleus to be detected on a screen. We would like to show that the probability density comes out correctly. To do this, we used a model of the detector screen consisting of a collection of bare protons held at fixed random positions. The electron, upon encountering the protons will radiate and eventually come to rest on one of the charge centers. This process involves the emission of radiation as the electron's energy is released during the binding process. The electromagnetic cascade contains precisely the information as to which proton finally accepts the electron. Time reversibility requires that the radiation, if reversed, would bring he electron back to the original wave-function. The detection of the particle is a statistical process if the information carried away by the radiation is lost to the observer. The selection of a proton is random and depends on the relative outcomes of the spontaneous emission process. Because the transition rate is always proportional to the square of the initial wave-function, the density, as observed by the final location of the electron, follows the original probability density wave. No point classical events are involved and no formal measurement processes is required. The idealization of this process gives the probability density, in the usual form, as the square of the absolute wave-function.
Null interactions
A correlated state of two particles, such as a spin singlet, is maintained by an entanglement composed of forward and backward null interactions. All bound states are of this form. Quantum computing is based on such forward and backward null interactions. Because a change of the quantum state may require more than one photon for completion, correlated emission or absorption may be observable.
Edge of the universe (optional)
New issues arise with these assumptions. Consider a flashlight at the edge of the universe, pointed out into nothingness. The switch is turned on. Will light be emitted into a region where no absorbers exist? Yes or no? The mechanistic theory requires symmetry in interaction, hence the light cannot escape; there are no particles to provide the advanced fields that could remove the energy from the source. The filament will increase in temperature and may burn out. The alternate possibility that the light my be emitted entails failure of conservation of mass, energy, momentum, and information. In essence, this issue is related to the information paradox in black hole theory. As the universe expands, the number of particles in the forward light cone will decrease, eventually guaranteeing a reduction in absorptivity. At present, there are no observations of reduced absorber efficiency for photons.
Super-luminal transmission 1 (optional)
If there exists such a limitation on absorber efficiency, it appears that spacelike transmission of data may be possible. Here is an example. A flashlight points through a transparent globe at a region of deficient reduced absorption. The sphere contains a gas that can be made optically opaque by exposure to light. A burst of radiation from a separated position will opacify the intermediate absorber. The flashlight emission will increase and the filament temperature will decrease. A drop in resistance is (?) initiated by the radiation burst.
Super-luminal transmission 2 (optional)
The space time diagram looks like this. The burst of radiation goes forward in time. The opacification retrodicts to the flashlight. The availability of reduced ultimate absorption is a prerequisite. The forward transition goes through without difficulty. The process does not violate relativity as long as the reverse signals are on null lines.
Kaon decay (optional)
No observational tests appear to be possible for photons but other null fields may provide better examples. Issues for gravitons remain open. But neutrinos also are fundamentally null particles. Data which shows retrodiction of information may already be available even though unanalysed. A decay of a PC invariant state into a preponderance of neutrinos or anti-neutrinos indicates, of itself, a violation of PC invariance. Differential absorption of opposite parity particles may be caused by the different absorption coefficients of matter. To lowest order, the neutron to proton ratio of the absorber implies a possible PC asymmetry. The effects of asymmetric advanced fields cannot be discounted theoretically and may at least be a confounding effect for this process. While the geometrical theories are fully parity violating, PC is an exact invariance. This mechanism provides a means to explain these experiments in a theory of presumed geometrical PC symmetry and time symmetry.
Kaon absorber effects(optional)
Experimental tests might involve changing the background absorber. Sensitivities are hard to predict, but known neutron to proton ratios do vary as much as ten percent for short distances. An alternative approach is to look for PC violating effects with confined experiments that do not emit neutrinos.
The cat
The cat paradox of Schrodinger's famous discussion is simple to analyze in this framework. The radioactive particle may in time find a free absorber for the neutrino that must be emitted. The causal chain will be complete and the cat will then die. There are no state superpositions. You can open the box to see.
Mental processes
The operation of more complex pieces of apparatus is of some interest. The von Neumann type 1 process is rejected. The identified mental recognition of a measurement process is no longer as compelling as it was in the early days. The combined classification of living things, such as animals, with inanimate objects, such as computers seems much more reasonable now. For a complex device, statistical variations in output can be due to deficiencies of the processing mechanism or failures in the incoming information. These effects are pseudo-random. Addition unavoidable variations come from the retrodicted effects of the advanced fields. These are truly random.
Summary
The important ideas that I have tried to present are
the availability of a
Fundamental mechanistic geometry,
the use of
Time-symmetric quantum interactions,
Spontaneous emission identified with advanced fields,
the
Randomness of measured density from electromagnetic cascade
the
Experimental test of statistics: God or mother nature.
the use of
Mechanistic processes for person, cat or atom.
that
Systematic prediction, random retrodiction determine mental processes.
And finally, I would like to thank Andrei and Ekaterina for their efforts to make this meeting interesting and useful.