Slide 1 Buenos dias. I am pleased to be here in Monterry, not just because the weather is better than up north, but because I get to talk to you about physics. Dr. Espinosa has suggested that I tell you about field theory, and I am happy to do that. I will start with some general comments about fields and then continue into the geometrical theories that I know best. Most of what I will show you comes from a long term study that began about 1970, and that was directed at finding fundamental geometrical descriptions of elementary quantum theory. Over time, more complex geometries were incorporated. The mathematics of geometry is well understood and has helped make sense of the difficulties of quantum theory. Using geometry for quantum mechanics has led to theories that integrate well with general relativity. This talk is intended to introduce these developments Slide 2 There are many types of field theory--classical and quantum in different combinations. Here are some questions to think about. Where does field theory come from, and how do we know if it is right? What is quantization? How does this fit with geometry? What is the right geometry? What in a geometrical system produces the quantum behavior? After that, since there are often questions about quantum theory itself, I will present an interpretation of quantum mechanics that is suited to geometrical descriptions. The wide variety of available interpretations must be left behind. Here, it will be worth stating a particular way of thinking that allows for a consistent language. Then, I will try to show you how a quantum gravitational theory looks on a particle by particle basis. The five dimensional theory, which has been studied in some form for nearly a century, has recently been shown to work for quantum wave fields. At this time, source, field, and motion equations are known and agree with quantum observations. They reduce correctly in the appropriate limits. Finally, if there is time, I will introduction a geometrical approach for electron spin. After that, there may be a few minutes to look at differences between the geometrical approach and current theories. A successful resolution should provide complementary information about the structure of physical laws. Slide 3 Electrodynamics is the model classical field. Real world fields are more complicated. The electrical cable is much simpler than the complex ice crystals that have accumulated on the tree Slide 4 Fields are abstract ideas used to study space and matter. The term wave-particles is a mathematical description of a particle that has wave properties, but that may have no representation as a point object. Points in space-time are used for the description but the objects themselves do not have a separate point-like existence. They are like particles only because they can be counted. The mathematical idea of a point concentration of mass or charge need not exist, even though it is inseparable from classical physics. There is not now nor has there ever been any experimental evidence for a finite concentration of mass or charge actually concentrated within a mathematical point. When carried out to sufficient accuracy, experimental tests always support the quantum theory and fail to agree with the classical model. Real objects occupy volume. Particle waves always appear, in any experiment. It is now well known that the electromagnetic field of a quantum charged particle comes from the whole wave function. A point object, in quantum mechanics, has come to mean a particle that has no internal structure. The word localization, has refers to the smallness of the space that the wave function occupies. In the end, Einstein's standard is most relevant. If you are thinking about the unknown, the issue is whether you can make things more understandable, the way they are. Descriptions may not become simpler. The test is see whether it is easier to understand what really is there. I use geometry for this because that is the way that I like to think. Slide 5 The concept of quantization must be modified to admit a quantum geometry. The usual process of quantization involves substituting operators into the classical Hamiltonian equation. A standard derivation of the Schrödinger equation is followed here. By substituting for the wave function and separating terms, comparison can be made with the classical starting point. The parameter beta connects the classical theory with the quantum. Zero for classical, one for quantum. The problem is that an intermediate value of beta agrees better with experiment than the classical value. In this case, we can feel certain that zero is the correct value, but in a more complicated situation, it is more difficult to be certain of the origin of the terms before starting to quantize. The problem is that there is no way to do this in general because an experiment will always show some quantum departures and may naturally produce a value of beta that includes some quantum effects. A particular classical theory might have some quantum terms inside. There is no experimental way to verify the classical starting point. Subsequent quantization can compound the error. A better way must be found to obtain the equations. It may be important to avoid classical physics as a fundamental starting point. Ultimately, quantum theories must be evaluated, alone, against experiment. Slide 6 This problem came to light during the study of certain types of geometrical theories. For some combinations of properties it was found that essential quantum terms appear automatically, without any explicit quantization procedure. How can this be?! Apparently, some geometries have quantum effects built in, from the beginning. It was noticed that some of the second order terms of the Riemann curvature tensor may be put into correspondence with the essential quantum terms. No one knows why this should be true, but ongoing studies show that the essential quantum terms can come from a properly chosen geometry. More disconcerting is the realization that accepted classical theories, like general relativity in particular, contain part of the quantum theory, even though they are represented as un-quantized. Substitutional quantization of such a theory leads to extra difficulties. Is there a way to get quantum theories without starting from classical physics? No systematic procedure is known, but there are some constructions that work. Slide 7 A basic problem in theoretical physics has been how to describe interactions when special relativity is included. Einstein faced this issue for gravity. The twin paradox is used for motivation. Two clocks travel along different paths and return to the same place. In a purely Lorentz theory, the clocks can start together, but cannot be compared later because the uniform motion sends them farther apart. One clock must experience an acceleration to be returned for comparison. Experiment, confirms that the moving clock has accumulated less time. General relativity agrees. The acceleration in the motion must be treated by using curvilinear coordinates. This motivates the development of G.R. Is this type of complex mathematical structure necessary for other interactions? A simple experiment cannot be done because the other forces are used to make the clock. However, similar mathematical inconsistencies appear for all interactions, and especially if quantum mechanics is included. All relativistic interactions should be studied with curvilinear systems. By convention, they are not. Most notable is particle physics, where relativistic motion is comon and accelerations are severe. A curvilinear system is ultimately required. The wave nature of the quantum particle and the curved lines of the probability current need a post-Einstein curvilinear treatment. Slide 8 The conformal transformation is the basic quantum element in a geometrical field theory. A conformal transformation can be characterized by the local effect it has on a small region surrounding a point. A rotation and a homogeneous expansion or dilatation are allowed. No other actions are permitted. In two dimensions it is easy to show that these transformations can be represented by complex number theory. The structure is much more c omplicated in higher dimensions because of the increased size of the rotation group. Conformal transformations are not intuitive. The matter that makes up the physical world has a certain characteristic property of size and structure. For quantum mechanics, it has come to be know that the conformal factors represent wave functions. I know of no exceptions. These are the quantum mechanical waves. The intuitive scale sizes of physical objects come from the length of the waves and not from the amount of dilatation or expansion that might occur. Thus the rigidity of material objects. Slide 9 Quantum mechanics is not easy to understand. Why is it like that? The elements that it uses are unintuitive and unfamiliar. In time we may come to understand it better. The real problem is that be try always to use classical physics, and God does not seem to know anything about it. In this context, let us begin a discussion of quantum interpretation. Slide 10 This is a model for a simple electron diffraction experiment. The electron moves through the slits as a wave and enters a detector. Imagine that the detector contains some positively charged particles, perhaps protons, that can capture an electron. After the electron enters the region, it cascades through a series of states, emitting light photon by photon. Eventually it comes to settle, as a wave, on one of the charged centers. The emitted photons characterize changes in the wave function as the particle cascades to the final bound state. The radiation contains a record of the state development. The appropriate concept is to associate the emission of a photon during the cascade as one step in a series of measurements that bring the electron wave to the final state. The collapse of the wave function is not instantaneous but proceeds only as fast as the speed of light. The non-relativistic interpretations of quantum mechanics suppose that the speed of light is infinite. The time required for evolution of the wave motion is mistakenly ignored. Slide 11 -- optional Particle descriptions of quantum probabilities are often justified from an experiment that uses a series of slits to refine the possible trajectories. One imagines that for some particle that finally arrives at the screen that the known position of each slit, in turn, implies a more precise knowledge about the position as it left the filament. The traversal of each slit entails emitted radiation and energy transfer to the atoms on the screen. Information about possible earlier trajectories is lost. Refinement is unsuccessful. The particle motion cannot be represented by a Lorentz transformed frame. The lines of probability current density are curved and density variations are significant. Diffraction alone requires a more complicated model. Slide 12 Consider two radio antennas placed close together so that the spacing is much less that one wavelength. Transmitting amplifiers are supplied that will cause a fixed amount of current to flow. The current ‘I’ in one tower will produce, at a given distance, a field ‘E’. When the second tower is turned on, the current is double, and the radiated field doubles too, but the energy increases by a factor of four. Where does the energy come from? The engineer points out that the voltage required to make the current flow is now twice what it was before. But why has this voltage increased. It is not directly from the other antenna, because the phase is wrong and the dependence on separation is not correct. The voltage is part of “the force of radiative reaction”. Is there a field that causes this force? Yes, one can be measured. By checking the energy conservation, for different currents in each antenna, it is easy to show that the radiative reaction field of one antenna is experienced by the other, in addition to its own. Where does the field come from? The effect has been know in atomic physics since the thirties and it has been shown that it can be calculated from the advanced field of the absorbing particles. In this sense, it is a real field (at least as real as a field can be), and it satisfies Maxwell’s equations with the absorbing particle as a source. Slide 13 These fields can be explained by making the radiative interaction symmetrical between the emitter and the absorber. That is, the fields emitted backwards by B affect the emitter A in exactly the same way that the fields emitted forward by A affect B. It is known that this process, assumed at the quantum level, can give a classical behavior that is that same as if retarded potentials were used from the beginning. For the geometrical theories, the time symmetric representation is preferred because the forces of radiative reaction cannot be added later by a phenomenological argument. They must be part of the fields and the geometry from the beginning. If this is not done, free fields must be added, and of a strength to cause the reaction forces. The sum fields become the same as if time symmetric interactions were used from the beginning. Without free electromagnetic fields, all interactions come form other charged particles. It is known that there are enough charged particles in the cosmos to insure that each possible photon can be absorbed. Experimental searches have not found any directions of reduced absorption. Slide 14 Of special importance in the interpretation of quantum mechanics is the existence of entangled states in which two particles are closely coupled as they move. The quantum current of each particle is distributed over an area of space-time. The currents may overlap. Each point of one current is connected, forward and backward, along a light like ray, to points of the other. This convolved form of interaction was called an entanglement by Schrödinger. The measurement of one particle always affects the other through the forces that go back and forth. All correlated states are of this form. Any one must be formed by some interaction, and all are fundamentally time symmetric. The measurement of one particle in an entangled state implies the separation of the particles sufficiently far to reduce the entanglement. These new lines connect to some other source which constitutes a new measurement for this second particle. This argument works for spin states as well. An interaction forms the specific state (i.e. a singlet), and a similar interaction must then break the entanglement. Slide 15 Correlation experiments compare photons emitted together by one atom. Sometimes an atomic transition cannot be completed unless two photons are emitted. There may not be an intermediate state that can be reached by one photon. The photons exit together and the sum of the radiative forces of each of the absorbers, return, as advanced fields to cause the atom to complete the transition to the final state. The measurement of one photon tells us something about the measurement result of the other--no matter which one happens first. In fact, a Lorentz transformation can change the order, so that it is not the same for different observers. How can this be? Apparently, the world connects each particle to the other, along light like lines. The internal workings must involve some deterministic mechanism. Slide 16 Schrödinger imagines that a cat is isolated in a box. A raditive source, may at some time emit a photon. Once triggered, the detector releases a vial of poison that kills the cat. Can the cat live in a superposition of alive and dead? I like cats as much as Schrödinger. But, the question is: is there determinism? Or is the result determined onlhy when the box is opened? The geometrical hypothesis is this: The radiative interactions of quantum particles are basically deterministic, as described by geometrical fields, and that all statistical effects come directly, or indirectly, from the collection of particles that provide the boundary conditions and serve to absorb (or emit) electromagnetic energy as the experiment progresses. The resolution of the paradox is that the cat is not isolated. It is possible to imagine a classical box, but only quantum boxes and quantum cats are available. The electromagnetic interaction extends to all particles. The possibilities of radiative emission must depend on the state of the absorbers, including the detector. Interactions may come from the past or future, or even through the box walls. The statistics cannot be kept out and a random observation of the emitted photon is certain but unknown. There is a particular outcome for the cat. Slide 17 That is enough of cats, now it is time for geometry. It is an assumption that there is a fundamental geometry for all of physics. This is supported informally by the practical usefulness of geometry in the world, and formally by the tests of equivalence. The point is to understand quantum mechanics as geometry. The identification of the conformal factor with the wave function does most of the work. The large number of particles, that make up our experience, makes the complexities of the quantum interpretation. Slide 18 In general relativity, the gravitational field is represented by one metric tensor, which is used to calculate the motion of all classical particles. These particles cannot overlap because they are point-like. The metric can give all the trajectories because each one uses a separate part of the metric. For quantum particles, there is no localized trajectory. A congruence or probability flow must be used and the motion of one particle can and will overlap another. The accelerations of particles, where they overlap, need not be the same. A single metric will not work in a region where different particles overlap. Multiple metrics, one for each particle, are required, and the universal gravitational metric must be defined in the classical limit by averaging over quantum states. Slide 19 The geometrical prescription is to use a separate tensor for each wave-particle. Essential quantum terms appear automatically. Any required fields, electrodynamic, gravitational or quantum are combined into a metric. One studies the invariants, usually the Riemann tensor, to find properties that characterize the physics. For quantum mechanics, five dimensions has been sufficient. Eight are needed for half integer spin. Slide 20 -- optional A number of problems from conventional field theory can be avoided. Theories should not have ghost solutions because of double quantization. Experimental tests should be applied alone without classical arguments. General relativity appears as a result and never as a starting point. An initial classical mechanics is not used. Full curvilinear descriptions avoids, at least in principle, the necessity of using straight segments between point interactions. Quantization is incorporated from the beginning. Slide 21 Derivations of five dimensional theory are beyond the scope of this talk. But, most people have not seen what a quantum gravitational theory might look like. So, I will show some equations and tell you about them. The initial construction is to incorporate the proper time as a fifth coordinate. It is placed on the right side and combined into a five dimensional metric. This leaves space for the electromagnetic vector potential. In the proper gauge, it becomes the velocity of the probability current and gives a precise connection with the quantum theory from the beginning. The intuitive notion of proper time is maintained by requiring all displacements of the motion to be null, i.e. of zero length. Slide 22 This null motion, when taken along geodesics, provides a line of probability current that can be projected onto space-time. The resulted projected line represents the sum of quantum, gravitational, and electrodynamic motion. The correct classical limit is obtained. (discuss details of slide) Slide 23 This null motion, when taken along geodesics, provides a line of probability current that can be projected onto space-time. The resulted projected line represents the sum of quantum, gravitational, and electrodynamic motion. The correct classical limit is obtained. (discuss details of slide) Slide 24 -- not required Here is an example. The lower picture is scanned from the book by Segre, while the upper one is from a paper by Stueckelberg. They both represent pair production. The experiment shows the particles emanating from a point, apparently with an acute angle at formation. The theoretical discussion portrays the event as happening smoothly, with the electron connected to the positron by a curved path. It should not be surprising that the curved representation is from the point of view of general relativity. An experiment to know what really happens at the vertex is difficult, and conventional Q.E.D represents the event with a sharp vertex. It is precisely this type of vertex that causes some of the renormalization problems. Slide 25 -- not required Fields available in the laboratory are much smaller that those observed inside atoms or during nuclear events. A calculation of the F-squared term gives the order of magnitude of possible effects. ( I will not guarantee the factor 3/16, it is hard to calculate and depends on the ultimate internal dimensionality of the geometry.) For a laboratory magnetic field of one tesla, the correction to the mass squared is .1 mev-squared. As a perturbation to the electron mass, this is two parts in ten to the 19, and is almost certainly unobservable. It is expected, however that this type of term should be present for neutrinos as well. A correction of this magnitude may be within reach of current experiments. The sign is negative, representing a kind of ‘neutrino diamagnetism’. This is a propagational correction, assuming a free rest mass of zero. Slide 26 The five dimensional construction generates source terms for the Einstein and Maxwell field equations. A five-conformal origin is assumed for the external interactions. If properly arranged, three source terms appear for the space-time gravitational Ricci tensor. The first is the usual mass energy contribution of the electrogmagnetic field. It appears in standard form and with a suitable numerical constant. The coefficient is small and has never been verified experimentally. The second term is of the classical form for “incoherent matter”. The quantum field of an individual particle contributes to the total source density. This term reduces to the usual classical limit if quantum effects are neglected. The magnitude of the velocity vector is not normalized for a quantum state and this may produce corrections to the gravitational field under extreme quantum conditions. The third term is a pressure. It is similar in form to the cosmological constant. In this construction it is a source term that does not contribute for classical particles. The factor in the numerator is zero in the classical limit. A factor of Plank's constant must be included, giving unmeasurable effects in a laboratory setting. A negative magnitude is possible (gravitational repulsion) and may be important in astrophysical or cosmological settings (black holes, jets, . . .) especially if the denominator can be made small. The Maxwell equation also appears in normal form except that the density is determined by the probability current of the quantum state. The fine structure constant appears between geometrical quantities and may eventually be calculable. Standard quantum electrodynamics is supported in a suitable limit. Slide 27 -- not required Overall, real cosmological effects are possible, and no calculations of the effects of a quantum gravitational theory are known. Some general comments are pertinent. The classical theory of the horizon may not apply, and displacements away from the classical position may occur, depending on the specific quantum wave function of the particles inside the ‘black hole’. Dipping of the horizon below the material surface may well produce ejection of hot matter, just because of the extreme thermal conditions. Astronomical ejection of matter is know, but not well explained. Different patterns may occur, depending on the size and rotational structure of the object. These theories have no apparent mechanism for information loss, so ejection by some means may be required. Gravitational pair production may occur as part of the process. Slide 28 -- optional Issues of second quantization are difficult. Electrodynamics is the model, but the assumptions required to predict the gravitational structure are not known. For a geometry, it is easiest to assume that electromagnetic and gravitational fields are always attached to particles and that emitted energy is extracted by the advanced field of the absorber. The tests of symmetry in electrodynamics are indefinite because it is too strong. Gravitation is too weak. Measurement of an asymmetry factor could be the most important result of gravitational wave astronomy. Gravity is non-linear, so unlike the electrodynamics, the use of symmetric fundamental fields gives a different predictions. In this type of geometrical theory, both the electrodynamic field and the gravitational field come from the same five scalar, and should have the same fundamental properties with respect to time symmetry. Slide 29 Spin has been an issue in the interpretation of quantum mechanics because of the statistical effects in correlated states. The conflicts are resolved by the geometrical interpretation as presented earlier in this talk. Now, a deeper question is addressed. Is there a fundamental geometry for spin? Slide 30 I am going to give a few details now for those of you who are familiar with spin theory. The Dirac equation can be converted to five-dimensional form by using the fifth anticommuting matrix. A very simple equation results. An extended space-time gradient of the wave spinor is dotted and simultaneously multiplied into the gamma matrices. The combination of spinor and vector quantities is rather peculiar to the Dirac equation, and has no precedent. There is something new to understand here. Slide 31 These five matrices are connected to the five metric with anti-commutation relations exactly as in four dimensions. Slide 32 The geometrical construction is this. Eight real coordinates are combined into four complex pairs and their conjugates. This is done to give a recognizable final form. Local Lorentz transformations are assumed to act on these complex four vectors according to the spinor laws. A local linear relation is assumed between displacements in the spinor space and in five space. The Dirac matrices are used to insure that the five vectors have the correct Lorentz behavior. The sum of complex conjugate terms insures that the five displacements remain real. An arbitrary spinor, zeta, parameterizes the relative local orientation of the spin space to the five space Slide 33 Assume simply that the eight dimensional system is conformally flat. The condition that the scalar curvature is zero can be written as a d’Alembertian on a wave parameter Psi, where the conformal factor is Psi to the 2/3 power. This insures a fundamental geometrical basis. Slide 34 The Dirac wave function is set equal to the eight-gradient of the wave scalar. The d’Alembertian becomes a spinor divergence on this vector, using the spinor metric. If the divergence is expanded according to the chain rule, as derived from the assumed coordinate relation, one gets, after simplification, the spinor zeta, multiplying into a spinor gradient expression that is equal to the Dirac equation. The Dirac equation thus characterizes the properties of a zero scalar curvature in a conformally flat space. The solutions look like electrons Slide 35 The assumption of a gradient structure for the spinor wave function indicates that local solutions of the Dirac equation can be found by differentiation from five dimension Klein-Gordon wave functions. This is verified by direct calculation. A solution, derived in this way does look like a spinning electron. Slide 36 -- quick look For dyed-in-the-wool geometers, application of the Pluecker-Klein correspondence provides some immediate useful information. Various combinations of spinors satisfy algebraic identities. These can be applied algebraically, even though they discovered in studies of projective geometry. Slide 37 The invariants correspond, in the classical limit to the energy momentum condition for a free particle. Slide 38 For a mass zero particle, the sum of the A-4 and A-5 squared must be zero. This, in turn implies that the particle must either be a neutrino or anti-neutrino. The same eight dimensional condition on the conformal curvature produces an equation for the neutrino. It must be possible to view the neutrino as a transformation of the electron by suitable rotations in eight space. It is a mechanical representation of weak isospin. An automatic connection to weak interaction effects is implied by this geometrical representation. Equivalence requires it. If an equation correctly describes an interaction, then this interaction must be equivalent to all others. For geometrical physics, this is a interesting as it gets. Slide 39 Here is a summary of various theories that attempt to combine different interactions. The standard model includes electrodynamic, quantum and weak interactions and is augmented by quantum chromodynamics to make the most complete theory of particle physics known. The five dimensional theory includes gravity and goes through quantum mechanics and electrodynamics exclusive of spin. The eight dimensional theory, still being developed, apparently extends into weak interactions. It is all still unknown what might lie ahead. Slide 40 Liking, as I do, to try to solve these puzzles, I am happy to have come to understand some more about how the world works. I am also happy to have a few more things to think about. There are a number of issues where this simple geometrical point of view comes up against other ideas. I believe in this approach, and I think that the differences will be resolved. Slide 41 Here are some of the more interesting questions. Most importantly is the beginnings of a theory of mass. No other approach is known. But, the combination of the quantum mass with the gravitational mass is very encouraging. The usual construction of mass with the Higgs mechanism is not obviously compatible. The concept of inertial in gravity theory is strong and a detailed construction is needed. The curvilinear description of elementary particles is long overdue, and there is much to be done. Segments of straight motion with sharp corners may have limited usefulness. A deep understand of particle transmutation may not be possible without the geometrical formalism. Regularization requirements may be resolved, and this, without affecting the know results of renormalization. The vacuum should really be understood as a physical effect from other particles. Finally, who knows where black holes will lead us? Slide 42 This is a lot of stuff. I hope that it has all been interesting for you. Working on it has been interesting for me. I have tried to give you some idea of a purely geometrical theory of physics. I started with a few basic concepts, then talked about the interpretation of quantum mechanics. Then, an overview of the workings of a quantum gravitational theory. I hope that a short discussion of spin will convince you of the value of this way of thinking. There are many things yet to study, and I have listed only a few. Slide 43 Finally, when the weather gets cold, its time to stay inside and think about physics. I do appreciate the opportunity to get away to a warm place for a little while. And I appreciate the chance to come here to give this talk. I thank you all.