By Derrick Crawford,2014-12-19 03:42
7 views 0

The Implications of Peter Lynds 'Time and Classical and Quantum

    Mechanics: Indeterminacy vs Discontinuity' for Mathematical Modeling"Professor Philip V. Fellman, Southern New Hampshire University

    Maurice Passman

    Professor Jonathan Vos Post, Woodbury University, Burbank, California Andrew Carmichael Post, California State University, Los Angeles

    Professor Christine Carmichael, Woodbury University, California

Draft 1.0 of 3 June 2004, approximately 3400 words

    [submitted for Proceedings, North American Association for Computation in the Social and

    Organizational Sciences, 2004]


    This paper examines the implications of Peter Lynds 'Time and Classical and Quantum Mechanics: Indeterminacy vs Discontinuity' as supported by thought experiments and possible physical experiments as suggested by Post, Post, and Carmichael. The major conclusion is that Lynds is correct in dismissing “imaginary time” as postulated by Hawking, although “imaginary mass” and “imaginary momentum” do not appear to be superseded on a theoretical basis. Generalizations may be applicable to mathematical modeling in the Social and Organizational Sciences.


    In Peter Lynds controversial paper 'Time and Classical and Quantum Mechanics: Indeterminacy vs Discontinuity' [Lynds 2003] he intriguingly argues:

    “It is postulated that there is not a precise static instant in time underlying a

    dynamical physical process at which the relative position of a body in relative

    motion or a specific physical magnitude would theoretically be precisely


    This postulate is obviously important at the very foundations of mathematical modeling and computation in both the Physical Sciences and the Social and Organizational Sciences. Rather than assuming the conventional structure of static moments in time at which the dynamical process being modeled is taken to have a precisely determined set of


    magnitudes, and then computing in discrete time (difference equations) what is contradictorily said to be really in continuous time (differential equations), Lynds states simply but counterintuitively [p.343]:

    “Time enters mechanics as a measure of interval, relative to the clock completing

    the measurement. Conversely, though it is generally not realized, in all cases a

    time value indicates an interval of time, rather than a precise instant… Regardless

    of how small and accurate the value is made however, it cannot indicate a precise

    static instant in time at which a value would theoretically be precisely determined,

    because there is not a precise static instant in time underlying a dynamical

    physical process. If there were, all physical continuity, including motion and

    variation in all physical magnitudes would not be possible, as they would be

    frozen static at that precise instant, remaining that way….”

    Peter Lynds, giving what to these authors is a compelling explanation as to why this is the correct solution to those motion and infinity paradoxes of the ancient Greek mathematician Zeno of Elea.

Lynds‟ analysis leads to his conclusions, including:

    (1) A body (micro and macroscopic) in relative motion does not have a precisely

    determined relative position at any time, and all physical magnitudes are not

    precisely determined at any time, although with the parameter and boundary

    of their respective position and magnitude being determinable up to the limits

    of possible measurement as stated by the general quantum hypothesis and

    Heisenberg uncertainty principle [Wheeler 1983], but with this indeterminacy

    in precise value not being a consequence of … quantum uncertainty….”

    (2) Imaginary time “is not compatible with a consistent physical description…”

    (3) “Lastly, „chronons,‟ proposed particles of indivisible intervals of time, also

    appear to be superseded on a theoretical basis, as their possible existence is

    incompatible with the simple conclusion that the very reason physical

    continuity is possible … is due to there not being a quantum or atom of time.”

    Lynds demolishes the cosmological proposal of “imaginary time” (and “no boundary condition”) as postulated by Hawking [Davies 1995], [Hawking 1997], [Hawking 1993], [Hartle 1983], [Yulsman 1999]. Post, Post, and Carmichael agree with Lynds, for reasons explained herein, yet suggest [Post 2004] “imaginary mass” and “imaginary momentum” do not appear to be superseded on a theoretical basis.

Peter Lynds summarizes the position that he refutes as follows:

    “Detailed calculations have been completed in the theoretical field of quantum

    cosmology in an attempt to elucidate how time may have „emerged‟ and

    „congealed‟ out of the „quantum foam‟ and highly contorted space-time 31/2] just after the geometry‟s and chaotic conditions preceding Planck scale [Gh/c

    big bang (new inflationary model). More specifically, it has been tentatively


    hypothesized that it would require special initial quantum configurations for the

    „crystallization‟ of time and the emergence of macroscopic (non-quantum)

    phenomena to be possible.”

Lynds explains [p.352]:

    “As soon as there is any magnitude of space (as a property of mass-energy), you

    naturally get the time dimension by default. If there is no mass-energy, there is

    no space-time. Because the reason continuity is possible is due to there not being

    a physical instant and physical progression of time [Lynd‟s main conclusions], it

    is not necessary for time to „emerge‟ in the first instance. The more appropriate

    question remains: how mass-energy, and as such space-time, can emerge,

    simultaneously bringing continuity with it due to the absence of a physical instant

    and physical progression of time; i.e. temporality or continuity would only be

    required to emerge from possible initial quantum configurations, states or

    histories in which time were a physical quantity.”

Lynds then argues [p.352]:

    “This conclusion… illustrates that temporality wouldn‟t need to „emerge‟ at all,

    but would be present and naturally inherent in practically all initial quantum states

    and configurations, rather than a specific few, or special one, and regardless of

    how microscopic the scale.”

    This argument rests on Lynds profound foundational analysis. He corrects Hawking et al. by pointing out [p.353]:

    “It is the relative order of events that is relevant, not the direction of time itself. It

    is not possible to assert using a model of the universe that includes a description

    of the sum over histories or path integrals of the actual structure of space-time,

    that time goes in any direction, let alone at 90 degrees to real time or linear time

    and takes on some of the properties, or is identical to that of spatial dimensions at 23-43 cm, 10 s[econds], while still being bounded approximately Planck scale … 10

    by the big bang (or possible big crunch, in a now seemingly obsolete closed

    universe) singularities in real or linear time, but having no boundaries in

    imaginary time. Neither real nor imaginary time exist in a consistent physical

    description, as time does not go in any direction.”


    Post, Post, and Carmichael agree with Lynds, that imaginary time is contradictory. But that takes careful analysis, as there are inherent linguistic, epistemological, and ontological biases in present social and organizational imagination which prematurely and ad hoc reject the possibility of imaginary numbers ever being applicable to physical reality. As summarized in [Post 2004]:


    “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.”

    Beyond the limits of conventional imagination due linguistic and historical constraints,

    the claim is usually made that these limits are not of human origin, but inherent in the

    physical world that we attempt to measure and model:

    “We note that a well-established principle of Quantum Mechanics [Bohr, 1938] is

     all that we have no ability whatsoever to measure any imaginary anything

    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”). ; is normally, by definition, a complex function. ; is

    defined everywhere in space.”

    But whether or not we can measure anything imaginary, conventional theory allows for

    imaginary variables in physical systems as useful tools for arriving at the right answer,

    regardless of nonintuitive and philosophically puzzling foundations:

    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.‟”


    Post, Post, and Carmichael also argue that one should not a priori dismiss the possibility of stranger values than the imaginary, for physical magnitude, including Quaternions [Raetz, 2000], [Rastall, 1964], [Adler, 1995]; Octonions; Clifford Algebras [Clifford, 1873], [Clifford, 1878], [Hestenes, 1967]; or Duplex Numbers [Kocik, 1999].

     suggest that “imaginary mass” (at low velocity, as Yet Post, Post, and Carmichael

    opposed to faster-than-light achyons) and “imaginary momentum” apparently violate no

    laws of Quantum Mechanics, Special Relativity, or General Relativity, and are a natural outcome of some String Theory and Brane Theory models [Hockney 2003]

    Specific, and experimentally testable predictions are made by Post, Post, and Carmichael [Post 2004]:

     -8Prediction #1: an event of at least 10 Joules = 0.1 erg = 100 GeV might create an

    imaginary mass particle: “imaginon.”

    This is the same energy range about to be achieved in the international search for the Higgs Boson, hypothesized to be the origin of mass for all particles of matter.

    Prediction #2: Imaginons, by gravitational interaction with other particles, will experience imaginary force.

    By Newtonian, quasi-Newtonian, and Relativistic derivations, a particle of imaginary mass, acted upon by a real force; or a particle of real mass acted on by an imaginary force; must experience an imaginary acceleration.

    Prediction #3: an imaginon acted upon by a real force accelerates in a direction orthogonal to normal 4-space.

    The 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 Quantum Mechanics, for very small distances and times. Historically, when Pauli proposed the neutrino to explain an apparent violation, Born offered that for subatomic scales, maybe sometimes there could be a violation [Born, 1924].

    Post, Post, and Carmichael predict a genuine violation of conservation of momentum IF a 100+ GeV collision creates an imaginon. The authors are 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, imaginons 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 32 kelvin, the Planck temperature), that imaginon creation would be (probably 1.4 × 10

    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.

    But how long would it take for an imaginon to apparently disappear, while leaving our brane? Certainly, by Lynds, it cannot happen in a single instant no such instant exists.

     -44“This process would take at least one Planck Time 5.4 × 10 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 -35higher) dimensions. That thickness may well be one Planck Length 1.6 × 10

    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.”

    Let us clearly relate Lynds‟ analysis with that of [Post 2004]. First, what is the reality of negative distance? One might ask even an elementary algebra class what the length of the side of a geometrical square must be for the square to have an area of 9 square inches. The students immediately assert that the square must be 3 inches by 3 inches. If the teacher solves the quadratic equation and says: “how about –3 inches times –3 inches?”

    the students will almost always insist (correctly) that negative 3 inches is “unphysical” –

    and the brighter student will prove this by the definition that length is always an absolute value corresponding to an interval of space, and thus no less than zero.

Lynds explains [p.353]:

    “It is the relative order of events that is relevant, not the direction of time itself.

    The order of a sequence of events can take place in either one order relative to its

    reverse order, or in the reverse order, relative to the first. It is not possible for the

    order of a sequence of events to be imaginary in the mathematical sense as it is

    logically contradictory and meaningless to describe the sequence of events as

    being at right angles relative to another sequence of events. The opposite of this

    couuld [sic] be posited for the relative special direction of events, but events take

    place at right angles relative to each others on a regular basis, and that has nothing

    to do with their direction, or the direction of time becoming imaginary.”


    So what would it mean for a body to move with an imaginary momentum? Since we (in Newtonian Physics) take momentum to be mass x velocity, the only way to have an imaginary product is for either mass or velocity to be imaginary. [Post 2004] predicts that imaginary mass is possible. What is the alternative? It would have to be that velocity is imaginary. What would that mean? There are logically two possibilities:

    (1) an imaginary distance is moved in a real interval of time; or

    (2) a real distance is moved in an imaginary interval of time.

    Conventional Physics excludes the notion of imaginary distance. Yet that is the basis for the prediction of leaving the brane, as an imaginary distance cannot be along either x, y, z or t coordinates. Conversely, Lynds establishes that imaginary time is inconsistent, and absurd, notwithstanding the cleverness of the presentation by Hawking.

Lynds concludes [p.353]:

    “The fact that imaginary numbers appear when computing space-time intervals

    and path integrals does not facilitate that when multiplied by i, that time intervals

    become basically identical to dimensions of space. Imaginary numbers show up

    in spacetime intervals when space and time separations are combined at near the

    speed of light, and special separations are small relative to time intervals. What

    this illustrates is that although space and time are interwoven in Minkowski

    spacetime, and time is the fourth dimension, time is not a special dimension; time

    is always time, and space is always space, as those i‟s keep showing us. There is

    always a difference. If there is any degree of space, regardless of how

    microscopic, there would appear to be inherent continuity, i.e. interval in time.”

    Post, Post, and Carmichael have suggested several possible experimental tests to detect imaginons, which Lynds‟ theory does not seem to eliminate, and to determine if there is or is not an antigravitational force between particles which formally have imaginary mass in the sense of being unstable with a short lifetime, where Quantum Mechanics gives a proportionality between the reciprocal of lifetime and the imaginary component of complex mass. These will be detailed in a forthcoming paper.

    In the context of mathematical modeling and computation in the Social and Organizational Sciences, we conclude that Lynds is correct, and that scientists should be extremely careful not to make philosophically and logically inconsistent assumptions regarding discrete versus continuous time. Existing papers in the literature of Social and Organizational Sciences might productively be reappraised according to the scrupulous standards demanded by Lynds.


S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University

    Press, 1995


Neils Bohr, Warsaw Lecture, 1938

Born, “Zur Quantummechanik”, 1924

W. K. Clifford, Proc. Lond. Math. Soc. 4 (1873) 381

W. K. Clifford, Am. J. Math 1 (1878) 350

P. C. W. Davies, About Time: Einstein’s Unfinished Revolution, London: Viking, 1995

J. B. Hartle and S. W. Hawking, “Wave function of the universe,” Phys. Rev. D 28 (1983)


S. W. Hawking, “Quantum Cosmology”, in 300 Years of Gravitation, S. W. Hawking and

    W. Israel, eds., Cambridge University Press, 1997

S. W. Hawking, Black Holes and Baby Universes, London: Bantam Press, 1993

D. Hestenes, J. Math. Phys. 8 (1967) 798-808

    Dr. George Hockney, Jet Propulsion Laboratory, NASA, October 2003, personal communication.

Jerzy Kocik (Dept. of Physics, U. Illinois at Urbana-Champaign), “Duplex Numbers,

    diffusion systems, and generalized quantum mechanics”, Intl. J. Theoret. Phys., 38, 2221-2230 (1999).

    Peter Lynds, 'Time and Classical and Quantum Mechanics: Indeterminacy vs Discontinuity', Foundations of Physics Letters, Vol.16, No.4, August 2003

Jonathan Vos Post, Andrew Carmichael Post, and Christine Carmichael, “Imaginary

    Mass, Momentum, and Acceleration: Physical or Nonphysical?”, Fifth International Conference on Complexity Science, Boston, 17-21 May 2004

George Raetz, “Quaternions and General Relativity”, <>

Peter Rastall, Quaternions in Relativity, Reviews of Modern Physics, July 1964, pp.820-


    Joe Sansonese , String Theory Discussion Form, 7 July 2003 []

John A. Wheeler and W. H. Zureck, eds, Quantum Theory and Measurement, Princeton

    University Press, 1983

T. Yulsman, “Give peas a chance”, Astronomy, September 1999, p.38





Report this document

For any questions or suggestions please email