Imaginary mass, force, acceleration, and momentum

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Draft 6.0-Short [for ICCS 2004] of 27 April 2004 [8-page version + 3 pp. Bibliography]
Jonathan Vos Post

Mathematics Department

Woodbury University

Burbank, California

Andrew Carmichael Post

California State University

Los Angeles, California
Christine M. Carmichael

Physics Department

Woodbury University

Burbank, California


This paper analyzes a possible emergent behavior of subatomic and astrophysical systems, which involves Complexity at four levels: (1) dynamic implications of assigning a Complex value to variables which, by tradition, were assumed real; (2) analysis of the related literature in Newtonian, Quantum Mechanical, Relativistic, and String Theory contexts, which have a social and conceptual complexity from their mutually different assumption; (3) the possibility of pattern formation shortly after the Big Bang, in high-energy events today, and in hypothetical dimensions beyond 4-D space-time; and (4) practical complexity in performing experimental tests of these hypotheses. This paper constitutes a preliminary discussion of a foundational question. Are imaginary mass, imaginary acceleration, imaginary force, and imaginary momentum under any conditions ever "Physical" (i.e. in principal observable by direct or indirect means) or "nonphysical" (i.e. theoretically amenable to calculation, but inherently unobservable in the real world)? The discussion begins by hypothesizing a particle or object of positive imaginary mass in a co-moving frame of reference, and considers some logical consequences. One unusual interpretation is that imaginary mass allows for objects to “disappear” from our ordinary space-time and “leave the brane” to go somewhere perpendicular to ordinary reality. The predictions in this paper are “far out” – even Science Fictional, yet they do not obviously violate Quantum Mechanics, Special Relativity, or General Relativity. They are in the broad context of the scientific literature. They may have both microphysical and macrophysical observability in the laboratory or cosmologically. We review the related literature on mass, in Quantum Mechanics and Special Relativity; return to a pseudo-Newtonian analysis; and then approach the complexity of modern theory and speculation.

Descartes famously defined: “matter is that which has mass and occupies space.” [Descartes]. Descarte’s thoughts on matter’s physical “extension” as its true basis for existence played a role in Newton’s thinking to refute Descartes on mass. Descartes also defined matter by extension, position, and motion through space. However, for him, size and mass are not clearly distinct concepts, nor are (scalar) speed and (vector) velocity.

Unfortunately, except for Einstein’s General Relativity and Higgs’ theory of mass, we have little more understanding of mass than did Descartes, and no convincing reason to exclude imaginary mass from consideration.

As Joe Sansonese has stated [Sansonese, 2003]: “Is it not possible that the utility of the complex numbers in physics is related to non-spaciotemporal aspects of physical law? … It may well be that mass states must be scaled by complex numbers. Historically, how were the categories of force, involving mass and, hence, dynamics, and motion (kinetics) wedded solely to ‘real numbers’…?” We note that the term “real number” was coined only after Gauss and Euler did their breakthrough work on “imaginary numbers” – perhaps as a psychological ploy to deny ontological significance to imaginary numbers, which (unlike zero, fractions or negative numbers) have no compelling pictorial or kinesthetic models in most human minds.

It was fruitful for De Broglie to suggest that particles have wave properties [DeBroglie, 1924; 1972]. The difficulties arise in defining the appropriate wave equations, and to meaningfully relate its mathematical solutions with experimental observations. Heisenberg [Heisenberg, 1924] proposed “Matrix Mechanics”, in which observable quantities such as mass and momentum were represented by infinite-dimensional matrices which did not commute (i.e., xp is not identical to px). This was useful for calculating transition probabilities, and thus relative intensities of spectral lines from atomic energy level transitions. But it was almost useless in learning what those energy levels should be, even for something as simple as a hydrogen atom. Most scientists were queasy about infinite-dimensional matrices somehow being necessary to explain things happening in three dimensions of space and one dimension of time.

That is to say, although Classical Mechanics uses the Hamiltonian as a function that describes the state of a mechanical system in terms of position and momentum variables (symplectic variables), in Quantum Mechanics, the physical state of a system is more broadly characterized as a vector in an abstract Hilbert space (infinite-dimensional), with physically observable quantities being identified with Hermitian operators acting on those vectors. The quantum Hamiltonian H is the observable corresponding to the total energy of the system. The eigenkets (eigenvectors) provide an orthonormal basis for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues. Since, by definition, H is a Hermitian operator, the energy is always a real number. Depending on the Hilbert space, the energy spectrum may be either discrete or continuous. The Hamiltonian generates the time evolution of quantum states, according to the Schrodinger equation, which takes the same form as the Hamilton-Jacobi equation, which is why we call H the Hamiltonian. But why should we trust an infinite-dimensional Hilbert space to be the best representation of our familiar universe? And is it merely mathematical convention that energy (or mass) must be a real number?
Schrodinger made things simpler and (for most) more intuitive [Schrodinger, 1926]. His proposed wave equation, however, introduced the conceptual quandary that solutions were complex-valued functions. Complex numbers are nicer than infinite-dimensional matrices for most people, but still weird. Schrodinger was accepted because his “Wave Mechanics” made the same predictions as did Heisenberg’s “Matrix Mechanics.”

Yet foundational problems were not eliminated. Born showed [Born, 1924] that the solution to Schrodinger’s wave equations could be interpreted as a non-observable object, which was utilized only as an intermediate step in calculating relative probabilities of the various allowable outcomes of a physical experiment. Bohr and Heisenberg ramified this philosophy into the “Copenhagen Interpretation” [Audi, 1973], [Jammer, 1966], [Cramer,1986]. Schrodinger never accepted this. Einstein never accepted this. These pioneers, and others, insisted that nature must have a more direct explanation.

Schrodinger’s theory is non-relativistic. Modern attempts to create relativistic wave mechanics have been fruitful (such as with Dirac’s correct prediction of antimatter). Such theories are usually called “Quantum Field Theory.” Problems remain. Imaginary mass is one way to uncover such problems. Some have said: “It is doubtful if an imaginary mass is physically meaningful.” Let us momentarily set aside our doubt.
We note that a well-established principle of Quantum Mechanics [Bohr, 1938] is that we have no ability whatsoever to measure any imaginary anything – all measurements are of real numbers, and complex wave functions are theoretical constructs that can sometimes help to explain the purely real measurements. That is, the wavefunction which describes a particle is usually denoted by the letter “psi” (pronounced “sigh”). Psi is normally, by definition, a complex function. Psi is defined everywhere in space.
Whatever state a particle is in, its wavefunction has a complex value at every point in the universe. Also by definition, the magnitude of psi squared gives the probability of finding the particle at some specific position. The wavefunction is unnormalized, however. It tells us the relative probabilities of the particle existing at two places, but tells us no absolute probability anywhere. By convention, we normalize psi by insisting that the total probability of finding the particle somewhere in space is 1 (certainty). Note that it is only by definition and convention that a particle is required to be somewhere in space. There is no physical or theoretical basis to exclude, a priori, that a particle can somehow be caused to vanish altogether from space and go somewhere else, outside the universe in which that wavefunction is everywhere defined.

A similar development in Quantum Mechanics allows us to determine particle momentum from the wavefunction psi. For constant “p” and i=square-root(-1) and Planck’s constant hbar, we define functions of the form psi = exponential(ipx/hbar)

as being the “basis states of momentum.” Again, by convention, we normalize an a priori unnormalized function by requiring that the total probability of finding the particle with some momentum is equal to one. As Felder states [Felder 2003]: “This fact, which follows directly from the properties of Fourier Transforms, is one of those cases where the math seems to almost magically do what it has to in order to give you the right answer.”

With all due respect to Harry Potter [Rowling], I am not comfortable with a flick of a magic wand to determine the physics of the real world. It is not compelling that momentum must always keep a particle within the 3-space universe (or, relatively speaking, for a 4-momentum to keep a particle in familiar spaciotemporal territory). Imaginary mass casts that “fact” in question. Specifically, it is not proven, but merely a postulate of Quantum Mechanics that [Felder 2003]: “when the operator for a particular quantity, acting on a wavefunction, produces that same wavefunction times a constant, then that wavefunction is a basis state for that quantity, and the constant is the value for that quantity.” This is usually mathematized by stating: “the eigenfunctions of an operator represent its basis state, and their eigenvalues are the corresponding values of the measurable quantity.” Our fundamental question can now be stated as: “can mass and momentum ever physically have imaginary eigenvalues?”

Imaginary variables in classical theory are also discussed as theoretical constructs. For example, an “imaginary frequency” electromagnetic wave is interpreted as a wave attenuating (being damped) in amplitude. “Imaginary Power” in an electrical circuit is interpreted as “reactive power.” In general, an imaginary variable in Physics introduces the possibility of cancellations (in the spacetime interval) or superposition probabilities (interference). As Bohr remarked [Bohr, 1938]: “Even the formalisms, which in both theories [QM and Special Relativity] within their scope offer adequate means of comprehending all conceivable experience, exhibit deep-going analogies. In fact, the astounding simplicity of the generalization of classical physical theories, which are obtained by the use of multidimensional [non-positive-definite] geometry and non-commutative algebra, respectively, rests in both cases essentially on the introduction of the conventional symbol SquareRoot(-1). The abstract character of the formalisms concerned is indeed, on closer examination, as typical of relativity theory as it is of quantum mechanics, and it is in this respect purely a matter of tradition of the former theory is considered as a completion of classical physics rather than as a first fundamental step in the thorough-going revision of our conceptual means of comparing observations, which the modern development of physics has forced upon us.”

This paper does not consider problems of rotation and angular momentum. One reason that we decline to consider rotation is that classical theory defines a particle’s intrinsic spin as about an axis with an absolute direction. Thus, the outcome of any measurement depends upon the difference between this absolute spin axis and the absolute measurement axis. Quantum theory, however, says that there are no absolute spin angles, but only relative spin angles. That is, the only meaningful angles are differences between two measurements, whose absolute values have no physical significance. This leads to the Einstein-Rosen-Podalsky class of experiments. Worse than that, the relationship between quantum angle measurements vary nonlinearly, so they cannot refer to any absolute direction (as all directions are relative). Bell’s Inequality deals with this. Hence the “relativity of angular reference frames” in Quantum Mechanics has similarities to the relativity of translational reference frames in Special Relativity. We note, in passing, that Quantum Mechanics therein involves “… generator of rotations in the spacelike plane orthogonal to the plane containing electron current and spin vectors… The unit imaginary in the Dirac equation is necessarily identified with electron spin….” [Hestenes, 1967].
“Missing Mass” [references] can perhaps be measured, and mass or energy apparently vanishing from a region of space-time may be taken as an indication that something is leaving that region, perhaps along another perpendicular axis [references].

This discussion would have been unpublishable through most of the 18th to 20th century, as foundationally inconsistent with the entire body of scientific literature. However, the proposal by the late Gerald Feinberg of Columbia University [Feinberg, 1966] was not only published, but sparked significant theoretical and experimental activity.

Tachyons are a putative class of particles which are able to travel faster than the speed of light (c). Tachyons were first proposed before the age of Einstein by physicist Arnold Sommerfeld, then later reconceived and and named by Gerald Feinberg of MIT [Feinberg, 1967] and independently by George Sudershan. The word tachyon derives from the Greek (tachus), meaning "speedy." As summarized by John G. Cramer, University of Washington [Cramer, 1992]:
"What are the physical implications of a particle with an imaginary rest mass? Gerald Feinberg of Columbia University has suggested hypothetical imaginary-mass particles which he has christened 'tachyons'. Tachyons are particles that always travel at velocities greater than the speed of light. Instead of speeding up when they are given more kinetic energy, they slow down so that their speed moves closer to the velocity of light from the high side as they become more energetic. Feinberg argued that since there are no physical laws forbidding the existence of tachyons, they may well exist and should be looked for. This has prompted a number of experimental searches for tachyons which, up to now, have produced no convincing evidence for their existence."
"Some theoretical support for the existence of tachyons, however, has come from superstring theories. These 'theories of everything' can predict the masses and other properties of fundamental particles. It has been found that some superstring theories predict a family of particles with a lowest-mass member that is 'tachyonic', in that it has a negative mass-squared. I should add that such predictions normally lead to the rejection of the theory as 'unphysical'."

As both Cramer and Feinberg are also known to have been aficiandos of Science Fiction (and Cramer a published novelist), it is not amiss to consider the Science Fictional treatments of this concept. This is particularly apt, as the concept of imaginary mass quickly leads to consideration of Science Fiction tropes including faster than light transportation, antigravity, and time travel. Feinberg may have been inspired at Columbia University by a Science Fiction story. According to Professor Gregory Benford of the University of California at Irvine [Benford, 2000]

“I discussed [the tachyon concept] with [Dr. Edward] Teller. He thought they were highly unlikely, and I agreed, but worked on them anyway [at Lawrence Livermore] out of sheer speculative interest. With Bill Newcomb and David Book I published in Physical Review a paper titled ‘The Tachyonic Antitelephone’. [Feinberg] edited a science fiction fanzine in high school with two other upstart Bronx High School students, Sheldon Glashow and Steven Weinberg -- who later won the Nobel prize for their theory which united the weak and electromagnetic forces…. He [Feinberg] told me years later that he had begun thinking about tachyons because he was inspired by James Blish's short story, ‘Beep.’ [Blish, 1954]. In it, a faster-than-light communicator plays a crucial role in a future society, but has an annoying final beep at the end of every message. The communicator necessarily allows sending of signals backward in time, even when that's not your intention. Eventually the characters discover that all future messages are compressed into that beep, so the future is known, more or less by accident. Feinberg had set out to see if such a gadget was theoretically possible.” Benford also wrote the definitive backwards-time-communications novel “Timescape” [Benford, 1980].

The late Isaac Asimov, for example, used the concept to "explain" FTL (Faster-than-light) spaceships in several of his novels [Asimov, 1989]. In his fiction "Nemesis" he states that there are two dual universes, one with "real" mass and the other with "imaginary" mass ("imaginary" as in the complex numbers). In one of the universes it is not possible to go faster than the light, and in the other one it is not possible to go slower than the light. Objects with mass = 0 are the frontier between the two universes. In this novel, both universes coexist in the same 4-dimensional space, but one cannot in any way be directly observed from the other. Professor Asimov somewhat sidesteps explanation of how an object with "real" mass can be transformed into an object with "imaginary" mass, thus accelerating it instantaneously to a velocity greater than that of light in a vacuum. Nonetheless, according to the Lorentz equations, such objects with imaginary mass don't seem to break any physical law.

So we return to the hypothetical in the underlying question: Why do we assume that rest-mass of a particle or object must be a real number? Why do we reject imaginary mass as "nonphysical?" For tachyons, we routinely assume imaginary rest-mass in order that mass at superluminal velocity is negative. But how about imaginary mass for objects actually at rest, or at small velocity?
This paper makes two bold predictions, one of which is experimentally testable by contemporary and near-future equipment, the other of which is more speculative, but may have observable cosmological implications. The detailed derivations of these predictions will presented in subsequent paper, and must be omitted here due to page count constraints.

The first prediction is that an event of at least 100 GeV might cause the creation of an imaginary mass particle, or “imaginon.” Further, that particle, by gravitational interaction with other particles, will experience imaginary force, and accelerate in a direction orthogonal to normal 4-space, thus disappearing from our observable cosmos (or from our brane). This process would take at least one Planck Time 5.4 × 10^(-44) seconds. How long depends on the speed of the imaginon (real magnitude of imaginary

velocity vector) and the “brane thickness” of normal 4-space along the 5th (or higher) dimensions. That thickness may well be one Planck Length 1.6 × 10^(-35) meters. Travel of one Planck Length in one Planck Time would mean that the imaginon’s speed is that of light, which is infeasible by Special Relativity for a non-zero imaginary mass. The imaginon thus either travels sub-luminally, the “thickness” of normal 4-space is more than one Planck Length, or the “disappearance” takes longer than one Planck Time. We shall return to the questions of observability of this process.

The second, hazier prediction, is that an entirely imaginary mass universe exists, with at least 3 spacial dimensions of its own, adjacent and/or orthogonal to normal 4-space. This is what Isaac Asimov has predicted (in the fictional context of Nemesis). However, Professor Asimov neglected to think through the properties of that Imaginary Cosmos.

In such a cosmos, all particles are of imaginary mass. Hence all pairs of particles have, as previously discussed, antigravitational repulsion from each other. Thus, at first blush, no large cosmological structures would be produced, i.e. no stars, no galaxies, no supergalaxies. However, oppositely charged imaginons can orbit each other in pairs, so long as the electromagnetic attraction exceeds the antigravitational repulsion. Various interactions between imaginon pairs are possible, including Bose-Einstein condensation. Hence imaginary mass universe cosmological structures may be possible after all. The equivalents of fission may be possible (if the Weak Force operates similarly), and fusion (if the Strong Force operates similarly). The imaginary universe would be, in some sense, dual to our 4-space, but neither identical nor opposite in behavior.

The creation of imaginons may only happen in pairs, for events of at least 200 GeV (or double whatever the minimum energy needed to create a single imaginon). If so, the pairs would be expected to be of equal and opposite charge, thus conserving charge. CPT symmetry would be expected to apply, with similar violations. Such pairs might always be bound, and isolated imaginons not possible, just as isolated quarks are unlikely. Compex-conjugate mass imaginon pairs have a far-field gravitational effect on real mass particles that is asymptotic to zero.

Specific numerical prediction regarding imaginary mass: as seen from our 4-space, a particle of imaginary mass leaving the brane appears as a violation of conservation of energy and conservation of momentum. That IS allowed by QM, for very small distances and times. By Heisenberg [derivation omitted] we find that the energy needed to kick the imaginon out of the brane is roughly 10-to-the(-8) Joules = a tenth of an erg =

100 GeV This is, not coincidently, the same order of magnitude as the predicted minimum energy of a Higgs boson; was exceeded by the LEP, and will be exceeded by CERN next year (or is it 2005?). [Bagger, 2003] Note that NO experiment to date has shown any violation of conservation of momentum. When Pauli proposed the neutrino to explain an apparent violation, Born offered that for subatomic scales, maybe sometimes there could be a violation. As to my suggestion for a super-sensitive Eotvos experiment, see: [Will 1993], [Will, 1998]

Returning to the issue of observability of an imaginon-creating event, there are several concerns.
(1) String Theory (going all the way back to Kaluza-Klein) suggests that electrical charge might be momentum around a loop of a compactified 5th dimension, and other quantum numbers might be similarly described. If so, imaginon creation might appear not as violation of conservation of momentum, but as violation of conservation of charge, or strangeness, or baryon number, or the like. Violation of charge conservation should be detectible.

(2) It would be difficult to distinguish between an imaginon forming, leaving the brane, and carrying away momentum, on the one hand, and the creation of a massive neutral particle (Higgs or otherwise). The fact that I have not discussed imaginon angular momentum adds uncertainty.

(3) Imaginon creation is thus hypothetically observable, but hard to distinguish from other phenomena. Thus, this theory clouds the interpretation of some “new physics” effects that might be produced by current and near-future accelerator/colliders.
I predict a genuine violation of conservation of momentum IF a 100+ GeV collision creates an imaginon. I'm not yet sure how to distinguish that from a neutral particle of the same mass, or a Higgs boson. But that's for experimentalists, who are already on payroll for those Higgs hunting efforts. Similarly they may be created in supernovae, hypernovae, black hole collisions; or ultra-high energy cosmic rays may also create imaginons or imaginon-pairs when they collide with interstellar gas, intergalactic gas, dust, our atmosphere, planets and stars, or photons. Thus a search for imaginon events may be conducted even if our accelerator energies are insufficient.

As a final note, the universe shortly after the Big Bang would have such high temperature (probably 1.4 × 10^32 kelvin, the Planck temperature), that imaginon creation would be frequent. This is true in particular before symmetry-breaking freezes out gravity, and to an extent still true but less frequent after gravity is distinguished from the other forces.

Thus, our early universe would be expected to bleed away energy and momentum off the brane. This could have measurable effect on inflation, the timing of symmetry-breaking, the rate of cooling as the cosmos expands, and on the distribution of density in the early universe that leads to today’s cosmological structure.
The predictions in this paper are “far out” – even Science Fictional, yet they do not obviously violate Quantum Mechanics, Special Relativity, or General Relativity. They are in the broad context of the scientific literature. They may have both microphysical and macrophysical observability in the laboratory or cosmologically.


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paper at 1998 SLAC Summer Institute.



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