From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.physics
Subject: A discrete model for quantum mechanics, was: Simple observation ...
Date: 2 Oct 92 17:57:27 GMT
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070

In article <1992Oct1.203520.12217@asl.dl.nec.com>, terry@aslws01.asl.dl.nec.com (Terry Bollinger) writes:
> In article <299@mtnmath.UUCP> paul@mtnmath.UUCP (Paul Budnik) writes:
> 
> > But my formalism makes the same predictions of quantum mechanics with
> > the exception of tests of Bell's inequality...
> 
> DUH?  Did I miss something?  If you have a formalism, I certainly don't
> recall seeing it here.  From my own very casual skimming through a few of
> your entries I still haven't even quite figured out whether you advocate
> hidden variables or not.  Please enlighten us with a couple of equations.
> 
> 				Cheers,
> 				Terry

I have been posting on this for a long time. Attached is a summary
of some earlier postings that explain my position more fully. I am afraid
the ideas are too primitive for many equations. It is easy to write down
a precise description of a model of the type I have in mind. Going from
that to macroscopic predictions would be extremely difficult. I do not
think these primitive ideas are enough to convince anybody, but they might
stimulate thinking and show how a radically differnt approach might
be developed. What I would like to convince people of is the importance
of being clear about what we do and do not understand. I also want raise
the level of interest in tests of Bell's results.

______

Einstein, wrote in a letter in 1954: "I consider
it quite possible that physics cannot be based on the field concept,
i. e. on continuous structures." Fredkin has done work on an approach 
for developing a discrete model for physics based on cellular automata.

My interest in this topic dates from the lates 60's when I was a graduate
student. It seemed natural to me that the discontinuities of quantum
mechanics might arise from discontinuities in space, time and fields.
In addition I felt there were philosophical reasons why discrete models would
be more satisfying than a continuous one (a topic for a different group.)

At the time I developed some ideas that I think might lead to a realistic
theory. The starting point was the observation that the relativistic
Schrodinger equation for a single particle is identical to the wave
equation except for the term involving rest mass. The idea was simple.
Provide a completely discrete approximation to the wave equation.
Do this by starting with a finite difference approximation. Modify the
difference equation to force the solution to always assume integer values.

The ideal goal of this scheme is to have a single scalar field and a single
difference equation that defines the evolution of the field values. All other
physical phenomena are to be derived from this basis.

The attempts to prove that quantum mechanics is complete with respect to
quantum indeterminacy have failed and led to the conclusion that quantum
mechanics is not local. There is a simple way to avoid violating
locality. One can postulate that there are two conditions that
precipitate a quantum collapse: 1) an observation and 2) a potential
violation of locality. There is no experimental evidence against
this hypothesis. If standard quantum mechanics is correct, vast
quantities of information are being transmitted arbitrarily large
distances (up to thousands of millions of light years) instantaneously.
Based on the principle Ockham's razor, spontaneous quantum collapse is more
likely than violating locality.

If quantum collapse is local, it has a space-time structure that is
subject to experimental manipulation. The wave function must transform
from its current state to a new state. At the start of the transition
experimental results between two particles in a singlet state will be
correlated as predicted by quantum mechanics. At the end of the transition
they will conform to the local hidden variables model of EPR.
The time, duration and shape of this transition at each point in space,
is the space-time structure of the collapse. This structure is not
needed in standard quantum mechanics, because collapse is assumed
to be instantaneous throughout all space. It is this assumption that
is the non-local aspect of quantum mechanics.

If you agree that the universe is probably local (or if a future
experiment shows that Bell's inequality is not violated), a new physical
model is needed to account for the structure of quantum collapse.
It is plausible that a model that could describe the structure of
collapse could also predict the outcome of experiments, not merely the
probabilities of all possible outcomes.

There is a mathematical property of all completely discrete, deterministic,
time reversible, non-divergent models, that suggests a mechanism for a
hidden structure for quantum mechanics. Consider a four
dimensional grid of discrete points that represents the space-time manifold.
Define some discrete valued (say integer valued) function on this
manifold. Define a time reversible rule that specifies the new value of each
point in the grid as a function of the values in a neighborhood of the same
spatial position in the previous N time steps. Define some initial
conditions and iterate the solution. If the result does not diverge it must
loop and this loop must include the initial conditions. It must loop because
of discreteness and determinism. There are only a finite number of states
in any finite set of points so eventually some sequence of N time steps
will be repeated. From this point on the solution will loop through
a fixed sequence of states. The loop must include the initial conditions
because of time reversibility. At any point one can reverse the order
of N states and continue the simulation. The result will unwind the history
of states all the way back to the initial conditions. Thus any loop must
include the initial conditions.

The property just described can be thought of as a general
conservation law. In effect everything is conserved if one integrates
over a long enough interval. Further, any rule that maps discrete
values to discrete values and approximates the dispersion property
of physical laws like the wave equation or Schrodinger's equation
must be nonlinear. This nonlinearity can generate complex structures
that vary dramatically over short time intervals. This suggests a
mechanism for generating quantum indeterminacy from an underlying
deterministic hidden structure well maintaining absolute conservation laws.

_____________
>   2) What is meant by 'diverge'? The values at a point may be 
> bounded, yet the system never repeat. Examples include 'gliders'
> in J. H. Conway's 'Game of Life' cellular automata and involve 
> coherent motion of a 'pattern' off to infinity.  A bounded size 
> pattern would obey your description, but does this require your 
> 'universe' to be  bounded (See number 1)

This is an important point. There are two kinds of divergence of
relevance here:

	1. For any finite initial condition the sum of the magnitude
	squared of all the field values in future states has a fixed
	upper bound.

	2. Condition 1 plus the requirement that the spatial distance
	between any two nonzero field values also has a fixed upper bound.

As you point out both conditions are needed for my proof that states
loop. Only the first condition is plausible for a realistic model of physical
reality.

If condition 1 is met and an initial disturbance breaks up into two or
more separate components or `gliders' that move apart they will each loop
individually through some repeated sequence of states. My original
argument applys to such a structure. The bounding box in which
the looping occurs will move with each individual glider.
This breaking apart can only be repeated a fixed number of times.
If it occurred repeatedly with no upper bound condition 1 would be violated.

Only condition 1 would be required of a possible physical model.
Any initial conditions will lead to either looping of a single
structure or looping within multiple structures that
move apart from each other.

> 
>   3) Why must the loop be reversible? Mappings can be chosen 
> that are not, because they are not one to one.  Perhaps the real
> universe's mapping is reversible, but is this important?

The loop must be reversible if it is to always include the initial
conditions. I find these structures of possible relevance to quantum
mechanics because they show how one might reconcile the absolute conservation
laws of physics with quantum indeterminacy. This is only true if the looping
includes the initial conditions.

For gliders the initial conditions of the individual
loops will be the conditions at the time the components were
first separated i. e. the first point in time when they
no longer influenced each other.

An initial disturbance breaking up into gliders is a possible model
for particle decay. The time when the structures were
completely independent would be the point where quantum collapse had
completed and EPR like arguments as opposed to the predictions
of quantum mechanics would agree with experimental results.
The requirement that looping includes the initial conditions implies
that for any decay event there is an inverse particle creation event.

>      O.K., Let's assume the discrete space-time-value model for a moment.
> One would like it to be TOE (a theory of everything). As such, all known
> "forces" and "particles" must be encorporated within it. The space-time grid
> size must be small enough (Planck length?) to encorporate all known models of
> scale size [actually this may not be true since we only need a system which
> produces states in the "observer", "device" that is self consistent with
> respect to our models of scale (do electron-electron experiments REALLY mean
> that an electron is a point particle or only that the results of the
> scattering behave as though it were a point particle?)]. It must also embody
> the galilean principle of relativity where structure is preserved under
> translation. I would also think the global topology of the grid would be
> significant for cosmological models. I assume that twisting, bending and
> distortion of the grid have no meaning and that it is only the connectivity
> of local points that matter. I would also assume that true singularities
> (particle or black-hole) could not exist in this structure (the grid point is
> the minimum "size").
>      I would be interested in hearing your comments on what is being modeled.
> [I am reminded of a discussion with Charles Misner at the University of
> Maryland where he thought at one point he had found a solution describing the
> electron within a form of General Relativity.].

Unfortunately I do not have a solution for anything. It is only an idea
for a class of models that have a several properties that suggest the
models might be physically significant.

The grid would be a manifold, i. e. a topology of points without a
distance function. The relativistic distance function would involve
not only these points but also the field values defined on them
(to account for general relativity) and the frame of reference (which
can be moving relative to the grid). In such a universe relativity
would only be a highly accurate approximation. The grid spacing, for
example, could not be Lorentz invariant.

The Plank length would probably be a good first guess of the order of
the size of the grid spacing. I do not have a single model in mind and thus
the grid spacing is one of several free parameters.

The wave nature of matter would be primary in this class of models. All
particles would be extended structures in space. Consider some finite
difference approximation to the wave equation that is modified to be
completely discretized. For example it maps integer values to integer
values. If one starts with some initial disturbance it will disperse
as predicted by the differential equation up to the point where
the integer values are no longer large enough to yield an accurate
approximation to the wave equation. Because of the nonlinearity of
the discretized difference equation, I suspect that one of things
that can happen is the disturbance will break up into individual
structures that are analogous to solitons. The round off error in
the discretized difference equation will allow for a minimum
gradient to exist without further dispertion. 

I think of such entities as stable dynamic structures. The particle
nature of reality would exist because a minimum
`energy' is needed to create such a structure. Because of this these
structures could only interact as units. That is they could pass
through each other unchanged or they would need to transform from
one type of structure to a different type of structure.
If two structure interacted then the position of this interaction
would be the focal point of the structure or structures that were
created. This is how the classical notion of a particle's position would
be defined. The degree to which it would be localized would depend on
the wave structure of the particles created (quantum uncertainty).

Black holes would not be true point structures, but could be
small relative to normal particles.

You are correct that twisting the grid would not be meaningful. The
grid is not embedded in some continuous space but is postulated as
the topological basis for space and time. One of my major motivations
for thinking about such models is doubting that space can be truly
continuous, i. e. that there can be an infinite number of positions
within a finite distance.

You are also correct that the goal is to create a theory of everything.
I have mentioned in a previous posting how one can create a single
discretized difference equation that has solutions that approximate
both the Schrodinger equation and the wave equation. This suggests
the possibility of a unified field theory in which a single
scalar field value accounts for both the quantum wave function and
electrodynamics. It is at least conceivable that the other fields
of physics would result from the nonlinear nature of this same discretized
difference equation. Similarly it is conceivable that the parameters
of particles result from the particular stable dynamic structures
that are solutions for a particular difference scheme.

>      It sounds as though something like a "pilot wave" for a "particle" is 
> what you are considering. That is, the "particle" is extended in space (via
> field values) and yet it is a "stable dynamic structure".

I would not use that description. It seems to imply that the particle and the
pilot wave are different entities.

>      I believe an alternative interpretation of exactly the same type of
> field structure is also possible, namely that there are no particles.
> That's right, no particles, only "cascades of events".  By cascade I mean
> the influence (field) at one level (atomic) gives rise (field-field
> interaction) to a new configuration (field) at a higher level (microscopic)
> which in turn causes (field-field interaction) the measuring device (human
> eye) to amplify (cascade) the event ("photon"-"rhodopsin" interaction)
> to impinge upon our consciousness (undefined).

Particles are needed for two aspects of physics. They are used to
represent field quantization and provide a hook on which
to attach the parameters of an interaction. Particles in the model I am
proposing serve no function other than those. Photons (and I would expect
any particles obeying Bose-Einstein statistics) do note exist 
as separate particles until the field values have become so dispersed
that they must break up into independent structures of a minimum energy
or until an interaction occurs that must involve the transformation of
this minimum energy from within the complete field structure. 
Particles obeying Fermi-Dirac statistics have more of
a permanent existence. However, in a multiple particle interaction
the evolution of field values might have several different
interpretations for what particles exist at a given moment. The
only thing that would *really* exist would be the field values. Their
interpretation as particles would only be unambiguous once the
particles had dispersed sufficiently to create a unique interpretation.

I believe this is similar to what you are suggesting.

>      The field at the sub-atomic level must have structure that in the 
> aggregate is revealed to us as "matter" BUT I'm now getting a sense that 
> this structure need not be "composed of" specific "particles" only that the
> interaction of the field with a part that is in a given configuration gives
> rise to other states that we call, say, scattered particles, excited states
> of atoms, etc.  The field interaction must simply be consistent so that
> every time an experimenter sets up a specific situation he gets similar
> results.

As described above in the model I am proposing there would be instances
where the indentity of individual particles would be clear and others where
they would be ambiguous or nonexistent.

>      By eliminating the "particle", the double slit experiment becomes
> readily understandable.  A situation is created (light source impinging upon
> mask with 2 slits) and the results observed (interference fringes on screen
> behind mask).  The field from the source passes in the classical wave manner
> through the slits and interfers with itself.  There is (are) no particle(s)
> passing through the slits.  The interaction of the field with the "matter" of
> the screen creates (in a probabilistic manner) events (cascades of events)
> in the screen.  The creation process is probabilistic since the dynamic
> state of the screen is highly complex and varying.  The right "phase" and
> "amplitude" conditions must be locally present to create an event.  The fact
> that decreasing the light "intensity" decreases the probability of an event
> is essentially a tautology.  One defines "light intensity" by the number of
> events created.

This is much as I would interpret this experiment. I would not say particles
are eliminated, but certainly the standard idea of wave-particle duality is
effectively eliminated. As you suggest there is only a continuous field
passing through the slits. The particle nature of an interaction comes
about because the field can only transform from one stable dynamic
structure into another stable dynamic structure. The conditions needed to
create an event would be complex because of the nonlinearity (the
discontinuous structure of space time and the field values). It would
depend on exactly what stable dynamic structures exist, how stable they
are etc. Chaos theory where nonliner systems can bifurcate into two
stable states is the closest analogue I can think of.

The last point about intensity is not how I would describe things.
If one has a high intensity source then one has a close approximation
to the classical continuous wave solution as far as the field values are
concerned. A larger amplitude of these field values simply increases the
probability of interaction. If the intensity is low enough one will reach
a minimum level for creating a stable structure. At this point you will
get a minimum energy structure emitted with a decreasing rate as opposed
to a less intense classical field.

> >I had a gut reaction against these speculations.  For a simple chap like
> >me it is hard not to believe that there are no particles.  I have put a
> >luminescent watch face (when these were legal) up to my eye in the dark
> >and seen the individual scintillations.  Of course that doesn't mean a
> >whole lot.  But what about tracks in bubble chambers?  What would happen
> >if you did a two-slit interference experiment in a bubble chamber?
> >
> 
> Consider the 2 - slit problem. What would you say is the implication of
> the existence of the *single* 'particle' interference pattern concerning
> the existence of 'particles' ?
> 
> 

I am not saying that particles do not exist, but that they are a
secondary phenomena. The goal is to reduce all of physics to a
single difference equation defining the evolution of discrete scalar
field values on a discrete grid. Particles result because there are
dynamically stable structures that this model generates.
These structures represent minimum energy units that have a stable
existence. When the field values are intense these structures do not
have an independent existence, but only show up as a quantum effects
when an interaction occurs or after the field has dispersed to
the point where the minimum energy structures start to become
spatially separated.

In the case of a two slit experiment with a single particle the
particle would be a stable dynamic structure that was to a very
high degree of accuracy exactly like that defined from the
Schrodinger equation. The interference pattern would be essentially
the same. The mechanism that determines if an interaction
occurs and how it is localized would be as described later in this
article.

Accounting for particle tracks in a bubble chamber is straightforward
at least at the intuitive level that these ideas are at. 
For simplicity I will call the incident particle a cosmic ray
althogh today it would more likely be from an accelerator.
The cosmic ray `particle' is an extended structure. If it is a particle
with rest mass, it has an intense core where the field amplitude is high
enough to be in the domain where the difference equation approximates
the relativistic Schrodinger equation. This would approximate to an
extremely high degree of accuracy the standard quantum mechanical model
of the wave function for the cosmic ray. It has an extended structure
where the amplitude is lower and the difference equation approximates
the wave equation. (Note in the case of zero rest mass particle
the relativistic Schrodinger equation and the wave equation are
indentical. Only this extended structure would exist and would
account for both fields.) This extended structure would account for any
electromagnetic and other fields associated with the particle.
Interaction of this structure with the contents of the bubble chamber
involves a series of transformations of the cosmic ray structure and the
bubble chamber particles (or structures) that it interacts with.

These interactions are a transition from one dynamic stable
structure to a different dynamic stable structure for both the cosmic
ray and the bubble chamber objects. The question of
which transformations occur is complex and is analogous to the
question of which path a nonlinear system takes at a bifurcation. 
Each of these interactions seem to occur at a spatial focal point of
the structures that result from the interaction. The degree to
which this point can be localized depends on the structure of the
interaction (quantum uncertainty) and is only loosely correlated with
the size of the interacting structures. There was not a unique event
at this focal point that caused the interaction. The interaction might
have started at this focal point and moved out like the formation of
a crystal from a seed. However, the interaction involves a
transformation of the entire structures and cannot be analyzed
as occurring at a single point in space-time. It will not be completed
successfully if the entire structure can be transformed in a
way that is consistent with the conservation laws. The degree to
which the track can be localized depends on the joint wave function
of the interacting objects just as it does in quantum mechanics,
but there was no point like particle that traversed this track.

Paul Budnik
Newsgroups: sci.physics