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\hyphenation{one-di-men-sion-al
di-men-sion-al %
wave-pack-et wave-pack-ets %
Klein-Gor-don}
\author{Hrvoje Hrgov\v ci\'c\\
{\em Department of Physics,}\\
{\em Massachusetts Institute of Technology,} \\
{\em Cambridge, Massachusetts 02139.}}
\address{}
\date{}
%
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\title{ A negative probability approach to quantum mechanics}
\maketitle
{\begin{center}
{\sl Abstract}
\rule{.0ex}{0ex}
\end{center}}
An interpretation of
multi-particle quantum mechanics that is completely local,
in a three-di\-men\-sion\-al space, is presented.
The model given to illustrate this
interpretation is best described as being an analogue
of the random-walk models used to explain phenomena
governed by the diffusion equation; in both cases,
the random motions of discrete particles on a lattice, when suitably
averaged, yield a behavior consistent with their respective
differential equations.
By building on previous work by Feynman and others,
seemingly nonlocal phenomena such as the
EPR paradox, can be explained by
ascribing a physical meaning to negative probabilities
and negative events, as elaborated in the paper.
The
model is faithful
in the sense that a universe run according to such a model would
be indistinguishable from our own.
%\begin{description}
%\item[1] Introduction
%\item[2] Requirements
%\item[3] Problems of the conventional interpretations
%\item[4] Preparation
% \begin{description}
%\item[4.1] Locational models %Positional?;Location-based?
%\item[4.2] Relativity of histories
%\item[4.3] Negative probabilities
% \begin{description}
%\item[4.3.1] Protocols
%\item[4.3.1] External paradoxes
% \end{description}
% \end{description}
%\item[5] The lattice wave
%\item[6] A closer comparison with other interpretations
%\item[7] The need for a localist interpretation
%\end{description}
%Every situation has a history
%whose accidental placement of
%fermions may differ, but
%is nevertheless such that
%on the average, within each
%situation, QM holds.
\vspace{.6\baselineskip}
\noindent{PACS numbers: 03.65.Bz}
\vspace{.5\baselineskip}
The interpretations of quantum mechanincs
familiar to most physicists
are incompatible with any dynamics that is strictly local
in a three-di\-men\-sion\-al configuration space.
This paper presents a model with an interpretation
that remove this incompatibility.
%In fact,
%the ingredients of such a model are well-established
%in the literature.
The mathematics and dynamics
of the model have been described in a previous
paper,$^1$
%\footnote{
%H.~Hrgov\v ci\'c, ``Random Walks and Traveling Waves of
%the $n$-Dimensional Wave and Klein-Gordon Equations'', preceding article.}
hereafter referred to as [TW].
The results presented there, in addition
to being interesting from a computational point of view,
can be used to construct a complete physical model of
quantum phenomena.
%, and therefore, of the unverse itself.
There are practical as well as purely philosophical reasons that
make such a model desirable. With it, one could in principle simulate
quantum mechanics completely with computational structures
that are in principle totally mechanistic, and inviolably local
(e.g., cellular automata, connection machines, etc.).
Though the model is in fact implemented on a discrete lattice with
a discrete time evolution, the macroscopic results
it yields are relativstically covariant.
Many reasons have been given for claiming that quantum mechanics
is fundamentally beyond the power of such local approaches. However, these
claims beg the
questions of what {\it can\/} be done with
such methods, and specifically, whether or not
one may, in spite of the obstacles, construct
a useful model of quantum phenomena.
\section{Requirements}
It is useful to mention in advance the features
which are required of the
model,
beginning with some terminology that
will make what follows clearer.
{\it External\/}
statements involving the model are those
that we as the potential implementors of the model
may make. {\it Internal \/} statements, comprising
a subset of external statements,
are those which can be made by an observer
who is {\it himself part of the model that is being simulated}.
For example, if we were to simulate on a computer a
universe consisting of
the well-known cat in a box \'a la Schr\"odinger,
the only allowable information the cat may access
comes from the measurements
she makes (using eyes, whiskers, etc.) while at the same time obeying
all the physical rules of this model universe. That is, she
only has access to internal
information. We who are outside this
universe watching it unfold before our eyes on the computer
console
may also discuss such external concepts as, the
structure of the
lattice on which model system evolves,
the absolute phase of a quantum wave, etc..
This is despite the fact that to an observer internal to the
system, such statements might be inherently
unobservable, even meaningless.
Proceeding now to the requirements for the
model,
it has already mentioned above
that all interactions are to be strictly local,
in a three-di\-men\-sion\-al configuration space.
It should be underscored that, given our desire
to work with mechanistic models, such a requirement is no mere
philosophical preference; it is rather
a practical necessity which must supersede
any unnecessary objections to its
implementation. The same can be said
of the requirement that the model is to be
implementable within a three-di\-men\-sion\-al configuration space.
As these are both requirements arising from our
own abilities and limitations,
we cannot expect that nature will
automatically accomodate them.
But if there does exist a model
that satisfies these very basic
%practical requirements,
needs, then
even if that model were to challenge our intuition is some other way,
we should still consider it a preferable alternative
%KEEP THE REST?
to models that do not.
%do not satisfy the requirements.
%(We have to believe in free will; we have no choice---Isaac Bashevis Singer)
Second, given that internal and external notions have
been distinguished, {\it it is
required that the model faithfully
give quantum mechanical results for
any measurements made internal to the
system.} The model will be satisfactory
if it can pass a Turing-like test. That is,
the model will be satisfactory
if the answers to any questions that an internal
observer asks of the system
(i.e. any measurement that he makes)
are consistent with his belief
that his surroundings obey the laws of
quantum mechanics.
Of course, the results may be made
by an external observer without
the tedium of explicitly constructing
an internal observer, but they must
be interpreted by rules internal to
the system, as would be done if
the internal observer were indeed present.
Third, to simplify any implementation,
the model should allow for the
presence of some classical objects,
i.e., objects whose positions and
momenta can in principle be exactly specifed, e.g., perfectly
reflecting surfaces, and other types of
potentials. Even if one desires ultimately to
simulate a universe in which all things obey the laws of quantum mechanics,
i.e., a universe without classical objects, and even
if relativistic and other limitations make their very existence problematic,
the
usefulness of such classical objects in many situations
makes them an desired feature of any
model.
%We will elaborate this constraint below.
Last, what has above implied will
be stated explicitly; that is, the
model should allow
observers. An
observer is simply a memory device
that allows records the number of
some specified events encountered some point in space.
It is true that there is more to perception than simply
the recording of numbers of events. Even so, it is this operation,
as crude as it may seem, that forms the foundation
of measurement, in that it is the means of transforming
physical situations into numerical quantities.
Although an observer
is internal to the model,
it is possible to conceptualize its
presence, and speak merely of
numbers that would have been
obtained had an observer been present.
In fact, as will be shown, it suffices
in many situations to forget about the exact nature of
the observer and to retain only
the collection point and the number
of encountered events.
%If we were to imagine that the
%universe in which we live to
In our own universe for example,
acceptable observers would be brain neurons
or groups thereof,
a photograph, a history book,
%%%%%%a Schrodinger cat-in-a box setup,
and
so on. Note that memory devices are
not necessarily classical objects; in fact, their quantum
states must obey the same rules as
the objects they are recording.
\section{Preparation}
%Kill this section and make the following two subsections into sections.
%\subsection{Locational Models}
%In order to better understand the interpretation
%we are proposing, we shall first
%elaborate some concepts used in
%other interpretations and contexts that are common to
%the models we shal employ.
\noindent{{\it Location dependence}}
\vspace{.6\baselineskip}
A notable feature of the model
is that, in simulating the behavior of
say, some large number of identical fermions,
all the information contained within the system
is to be derived from the {\it location\/} of
these fermions in space. This is in
fact a feature of the world around us;
all measurements of physical quantities are manifested
in terms of the locations of particles.$^2$
%\footnote{
%A paper by J.~S.~Bell,
%``Beables in Quantum Theory'', in B.~J.~Hiley and F.~David Peat (eds.),
%{\it Quantum Implications\/} (Routledge \& Kegan Paul, London, 1987), presents related considerations.}
In trying to construct a model of quantum phenomena, it
suffices to construct one for which, at all times, the location of
particles is consistent with quantum mechanics.
To elaborate, one can speak of properties
such as the the momentum and mass of particles
by anticipating or recalling
the location
of those particles with respect to
some corresponding measuring devices (diffraction
gratings, mass spectrometers, etc.)
In fact, the measuring devices are themselves
also described by the location
of their constituent matter (pointers of a meter, and so forth).
Even events involving photons of light remain forever
unobservable unless they somehow affect the locations
of material particles, by way of some photoelectric device or
at the very least, the photosensitive neurons in
someone's eyes. This is not to say
that momentum and velocity of particles have no meaning.
Rather, all these properties can be inferred from
from the the way they affect the location of those
particles (or the particles comprising the
measuring apparatuses).
Any model or interpretation
that is thus constructed,
will be called {\it locational}.
%Likewise, it is only the
%questions dealing with the location of
%particles that can be considered valid, so that
Likewise, all questions a physicist can hope to ask must allow themselves to be
phrased
in terms of the location of particles.
For the sake of convenience, it may be assumed that
particles can also be distinguishable as electrons, quarks, etc.,
and that a given observer or other measuring device registers only
when the particles encountered are electrons, for example.
Alternately, one may also suppose that
only one kind of particle, say an electron, is observable.
Properties concerning other particles can then be deduced
in terms of how they affect the locations of electrons.
In any case, the locational model is one
in which {\it all information about the system at any given time is
obtained from the position of the various types of particles
within that system (or any portion thereof).}
The determination of the presence of these particles
thus becomes the goal of the measurement procedure.
\vspace{\baselineskip}{\obeylines
\noindent{{\it The past, as dependent on the present}}\nopagebreak
\vspace{.6\baselineskip}\nopagebreak
}
%
Note that to say that our universe exhibits this locational feature is to declare that
{\it the past does not exist apart from the present.}
In other words, the past, or more precisely,
any accesible information concerning it, may
be visualized as simply a function of
the present. Everything that contains our
notions of the past, including
history books, neurons, light
reaching us from a star distant
in space and time---all of
these are themselves
describable as an arrangement of
matter particles matter {\it in the present.}
Furthermore, these particles are
subject to the same laws of physics as those whose
behavior they record.
Suppose that some super-observer was able to
alter
the positions and arrangement
of these objects in a way
that resulted in a totally different
picture of the past.
Or else, suppose the internal workings of our
universe were such that a super-observer
looking on its evolution saw that
the conceptions of the past varied
from (his) time to time, and from
one (internal) observer to the next.
Even so, we,
the internal members of the universe,
would continue to believe
in the correctness of quantum physics, and of
our own private histories, as long
as each of the histories was in accordance with
such a belief.
This argument is reminiscent of
a familiar argument of Poin\-ca\-r\'e and others, elucidating the relativity of
space.$^3$
%\footnote{
%H. Poincar\'e, {\it Science et M\'ethode\/} (Ernest Flammarion, Paris, 1912),
%Livre II, Chapitre I, p.~94.}
If some
su\-per-ob\-ser\-ver suddenly increased
the length of all objects (relative to himself)
in the universe by the
same factor (and compensatingly
scaled the proper coupling constants)
the inhabitants of the universe would be completely
unaware of the change.
It is true of course, that by introducing such
su\-per-ob\-ser\-vers into the discussion
one can justify almost any physical
model whatsoever and therefore
one typically applies Occam's razor and
supresses any mention of such outside phenomena.
However, in constructing, say, a computer model
of quantum mechanics, such a notion remains
useful; the super-observer is merely
the universe simulator, who
is not himself a prisoner of the simulation,
and who in principle does have the power to
make the changes considered above.
Furthermore, the proposed model
has the value of being able to compute the numerical
quantities of quantum mechanics.
These results
are obtained in a notrivial way, and not as
the machinations and of some super-observer.
%
%I REALLY WOULD LIKE ANOTHER SENTENCE HERE,SOMETHING
%BETTER THAN THE PREVIOUS SENTENCE.
%
Lastly, the model has the very useful property of being
able to be completely realized within a three-di\-men\-sion\-al
space, and with a completely local dynamics.
\vspace{\baselineskip}{\obeylines
\noindent{{\it Generalized probabilities}}\nopagebreak
\vspace{.6\baselineskip}\nopagebreak
}
%
It is at first
unclear how one might make use of these nonabsolute histories.
%or indeed why anyone would want to.
However, there already exists in the literature
a related concept, namely,
negative probabilities. The
introduction of generalized probabilities has long
been known to simplify simplify
phase-space computations in quantum mechanics.$^4$
%\footnote{
%M.~Hillery,
%R.~F.~O'Connell, M.~O.~Scully, and E.~P.~Wigner,
%{\it Physics Reports\/} (Review Section of Physics Letters) {\bf 106}, 121 (1984);
%P.~A.~M.~Dirac, {\it Reviews of Modern Physics}, {\bf 17}, 195 (1945);
%R.~P.~Feynman, `Negative Probability' in B.~J.~Hiley and F.~David Peat (eds.),
%{\it Quantum Implications\/} (Routledge \& Kegan Paul, London, 1987).}
However, these probabilities
have typically been treated as merely
formal constructs, without any physical
meaning.
In the context of this model, they
are endowed with an objective reality
in the quantum world without violating the
macroscopically derived notions of causality.
Just as negative numbers were originally seen as
debits on some ancient accountant's ledger signifying
the taking away or the reduction of profit,
% gained somewhere else,
negative events, and their associated negative probabilities, are also
like bookkeeping devices that ``erase'' the occurrence of some other event.
%KILL THE REST OF THIS PARAGRAPH.
However, on the microscopic level, and in terms of the model of [TW],
this erasure has a more absolute effect than
the accounting example, or indeed any classical analogue suggests. Because
the ``past'' cannot be separated from the devices that record it, an
event which is erased is one which, as far as internal
observers are concerned, in effect, has never existed.
Such negative events will remain invisible to
observers if (1) any macroscopic measurement
of events yields a positive number and if (2) the
observers are macroscopic beings who can only perceive
macroscopic phenomena.
This however, is precisely the case in quantum mechanics.
We, for our part, as beings internal to our own universe,
are ineluctably macroscopic.
The same is true of the devices we use,
even when they probe macroscopic phenomena.
Regardless of what we perceive, whether it is
the output of a photomultipler tube,
the click of a Geiger counter, or the
scintilla of light from a sodium iodide crystal, no matter how
closely it is related to the quantum realm,
it is a macroscopic event, otherwise we
would be unable to notice it.
Finally, note that every act of perception
on the part of human beings is itself a measurement;
even if we perceive the results or records
of past measurements, the perception of these events
is a measurement in its own right, subject to
the same protocols and conditions as all other measurements.
%WE ELABORATE THIS BELOW.
\section{Protocols}
%This section shows how to incorporate some
%of the above ideas into the model of quantum mechainics
%described in [TW]. As discussed in more detail there,
%the model's configuration space consists of a
%three-di\-men\-sion\-al square lattice evolving
%in discrete time steps;
%nevertheless, this manifestly
%nonrelativistic system results in
%relativistically covariant wave equations.
%[TW] also shows how to simulate and obtain measurements
%(i.e.~event counts) for
%wave functions of any number of particles.
%(For the purpses of this paper, it may be assumed that for multi-particle
%events, the first counting procedure described in [TW] is the one referred to here.)
%xxxxxxxNew Section to replace above two para.xxxxxxxxx
This section shows how to incorporate some
of the above ideas into the model of quantum mechainics
described in [TW]. To summarize some of the results there,
each single particle state is represented by a gas of
particles in a three-dimensional
orthonormal lattice. At each time step,
the particles execute a step along the links connecting
nearest neighboring points. A particle coming into a given
point will has the probability of producing several
particles traversing away from that same point. Each particle
has phase associated with it: positive, negative, positive
imaginary, and negative imaginary. The algebraic number of particles
in a given link is taken to be the complex number whose real part
is equal to the number of positive particles minus the number of
negative particles, and whose imaginary part is equal to the number
of positive imaginary particles minus the number of negative imaginary
particles. Annihilation of
oppositely phased particles in any link occurs, resulting in
particles of at most two phases per link.
The distributions of the
phased particles coming out from a lattice point
that are produced by incoming lattice particles are
such that
the expected number of particles at ${\bf x}$ at time $t$
is equal to the wave function corresponding to a
single particle state, provided the initial distribution
of particles likewise corresponds to the initial
wave functions. The method of initialization
is also given. In spite of the fact
that the equations considered in the paper were defined on a lattice,
in the limit of small spacings, the method yields
relativistically covariant solutions of wave equations; the wave and
Klein-Gordon equations were considered explicitly, but
the method given can be generalized to other wave equations as well.
By superposing two distinct, and independently initialized lattice,
one is able to define events as coincidences of the two sets
of particles (call them bra and ket particles) at any given point,
and thus obtain event counts corresponding to the absolute
square of the wavefunctions, even though the increments
in any such event count are in general, complex.
These results are extended to multi-particle
systems as well.
%xxxxxxEnd New Sectionxxxxxxx
\vspace{\baselineskip}{\obeylines
\noindent{{\it Observers and counters }}\nopagebreak
\vspace{.6\baselineskip}\nopagebreak
}
%
The first step in expanding the results of [TW] into an entire
interpretation is to go from speaking of {\it event counting} to
{\it event counters,} and to associate these constructs with
measurements and the observers, respectively.
(Note again that in what follows, the observer may be any inanimate
measuring device, not necessarily an intelligent being.)
%In this section, we wish to include the counter, at least conceputally,
%in the universe we are simulating. Every internal observer
%in the simulated universe will be an event counter;
%the
%process of observation then is the transformation of
%physical phenomena into numerical quantities,
%KILL THIS NEXT CLAUSE?
%i.e., into language. Although, as the previous
%paper has shown, external observers may make
%measurements at any point of the universe whatsoever,
%the measurement can be said to be made by an internal observer
%only insofar as there is (ar at least, in so far as there can be)
%some internal physical object
%that takes on the role of the observer.
%A similar notion is found in
%the study of electromagnetism; Gaussian surfaces
%are merely theoretical constructs, that may
%be postulated ad libidum, but they can
%correspond to physical surfaces as well.
%The full ramifications of this correspondence between
%the physical and conceptual observer, akin to
%the mind-body problem, is beyond the scope of this
%paper, although we shall need to take
%note of this correspondence should we
%wish to simulate situations in which
%the behavior, (or indeed the existence) of the observer depends
%on the outcome of what is being measured.
%(consider again the cat-in-the-box experiment.)
For the purposes of the interpretation, and for the sake of simplicity, regardless of the
nature of the observer,
the paradigm of the measuring device will be an idealized Geiger counter, one that also
contains a reading giving the number of events encountered since the last time
the instrument was turned on or reset.
The features of this device are very basic.
The counter's primary component is an observation point, or {\it probe,}
which is merely some preferred point in
space, that may change as a function of time, and at which
the event counting takes place. For mul\-ti-par\-ti\-cle event
counting or combinations thereof, the counter will contain
several such probes, whose locations and operations
are subject to the restrictions
mentioned in [TW].
It will be supposed that each probe may
be in either an ``on'' or an ``off'' state, and that when it is in the latter state,
its reading is unchanged by the presence of
particles. The purpose of this shuttering feature is that it allows
the recording events in some specific window of space and time; additionally,
this feature can serve to ensure that a particle whose presence
is measured as it passes through the probe point does not
meander back to that point again and again, resulting in overcounting.
Most real measuring devices have a built-in shutter mechanism, resulting
in a minimum ``dead time'' that must elapse before additional
particle encounters can be registered, although as
discussed below, incorporating such a feature into the devices we consider
entails considerable difficulty.
As mentioned previously, the counter will at any time have associated with it a
{\it reading}, some nonnegative integer, that
changes in accord with the particles encountered
at the probe point. A
nonnegative ``post-count'' reading may be obtained from
the usual event count (which is in general a complex integer) simply by retaining
only the positive real part of that reading (or zero
if the real part happens to be negative.)
Also, the counter will have a reset mechanism
by which the counter may at any time be set to zero.
Time intervals in between successive
reset operations will be called {\it runs;} the
spacetime path of the probe point will also be referred to as
the {\it integration path} corresponding to a given run.
For now, it will be assumed that
the decision of when to turn the various probes on and off and
the choice of an integration path is made in advance of the event
counting, so that it
does not depend on the outcome of the event count.
\vspace{\baselineskip}{\obeylines
\noindent{{\it Multiplicity of integration paths }}\nopagebreak
\vspace{.6\baselineskip}\nopagebreak
}
%
It is important to note that {\it any integration path is a valid one,}
provided that in the macroscopic limit it
corresponds to a space-time path that some physical measuring
device can take. (Also, as discussed in [TW], it is necessary
that even for idealized measurement processes to introduce
perturbations into the phenomenon being measured, as they do it the real world.) In spite of
the multiplicity of valid integration paths, and given the notion of negative events,
there can be great differences between
the pictures of what is happening in a given physical situation
if those pictures are obtained from different integration paths.
This is especially significant if the events involve devices such
as photomultiplier
tubes and the like, whose macroscopic behavior is sensitive to
microscopic phenomena. As long as the different pictures (the ``many worlds'', if you will,)
obtained
from each integration path are always
individually consistent with quantum mechanics, it does not
matter if they are not consistent with each other.
%This is to be
%expected in
%light of what has been said above.
%% concerning the nonabsoluteness of the past.
Henceforth, it will be assumed
that only electrons can affect the readings of the counters;
doing otherwise would needlessly complicate the presentation.
As mentioned previously, it
the positions and properties of other particles are to be
inferred from the behavior of the electrons that orbit
around, or are deflected by them, etc.
In spite of the simplicity of the event counting devices, they
can be used to obtain all the numerical answers one needs,
provided
that the surroundings are arranged in such a way that they ask the proper questions.
One can choose to sweep the probe past the output end of
a diffraction grating,
a mass spectrograph, a charged particle detector, indeed any classical device,
depending on what properties are to be simulated.
%We could have made the device even simpler by
%allowing each one to participate in only one run, thereby obviating
%the mention of resetting, but we have chosen to allow measuring devices
%to participate in many such runs.
\vspace{\baselineskip}{\obeylines
\noindent{{\it Memories}}\nopagebreak
\vspace{.6\baselineskip}\nopagebreak
}
%
It is very important to note that
these measurement have not been endowed with a memory.
%This fact will reappear again and again in what follows.
In order to
obtain the information that a memory device can provide, it is necessary
to make measurements
on these physical memory devices, or at least appeal to what such
physical memory devices would yield, had someone gone through
the trouble of actually simulating them. Furthermore, these measurements
on the records of past events must be distinguished from
measurements on the past events themselves.
%??KILL THIS PARAGRAPH
%In the above discussion of multiplicity of integration paths,
%where it was noted that internally speaking, the pictures
%that two observers may have of the world need not be consistent
%with one another, it may happen that the observation devices are
%programmed to record their observations and these
%records are then brought together to some common destination,
%in an attempt to manifest their mutual inconsistency.
%Given that these records are themselves physical objects,
%the best that one can expect is that a central observer,
%or even one of the initial two observer who comes to a central location,
%who makes measurements on the recording devices also
%arrives with some picture that is internally in accordance
%with quantum mechanics, and therefore, its macroscopic limit.
%However, to say that some central observer determining
%%according to him,
%the two records to be consistent with
%common sense, is not to say that an external observer,
%looking down on all three observers would find the
%pictures they imply are all mutually consistent.
%Thus, negative events and nonabsoluteness of histories
%become concepts that can be taken seriously
%once we realize that memories are themselves physical phenomena
%that must be observed in order to arrive at some conception
%of the past.
Of course, in constructing a model, we, or anyone else who
is external to to the model, will be able to
remember what
has happened, without regard of any internal memory device, but
such an external view of what is happening within the model will not, in general,
be at all like the views of the internal
observers.
%?????
There is a similarity between these conceptual measuring devices, determining
information in the model universe, and ourselves, observing the phenomena
in our own universe. This
becomes apparent if
we agree to consider the portions of our brains responsible for memory
as somehow separate, or auxiliary to ``ourselves'', so that these regions must themselves
be explicitly observed in order to obtain information from them.
When this is done, then we too, cannot claim to have an ``absolute''
knowledge of history. Instead, we must
always refer to (i.e.\ make measurements on) the various
portions of our brain responsible for memory, as well as
our lab notebooks, history books, etc., in order to
reference the past. Having done so, we can then
construct various correlation experiments among these
records of the past to
measure the consistency of their data.
As long as the results of these direct and correlation data
are macroscopically classical, we would have no reason to view the
world differently from the way in which we now view it.
Even if something as unusual as the erasing of previously encountered
events were occuring at a microscopic level, then so long
as the results we thereby obtained were nonetheless consistent with
quantum mechanics, and thereby its classical limit,
we might continue to think of negative events as impossible.
One way to ease the transition to thinking about negative events is to realize that, in the context of the model,
one can say that negative events, and for that matter, positive events, have
no meaning in themselves. Rather, they merely serve as instructions or
bookkeeping devices
that direct the changes of the counter. As far as internal
observers are concerned, no statements can be gained about the universe apart from
some counting procedure.
%It is then a trivial matter to introduce the notion of
%negative events. Encountering one means simply to
%decrement the counter instead of incrementing it.
And in terms of this counting procedure, the encounter of a negative event merely erases
the effect of a previous positive one.
As mentioned above, the counter reading may be made positive by fiat.
However, the laws of quantum mechanics,
as well as the law of large numbers, already ensure that any macroscopic
event count is positive, insofar as the net imaginary contributions of
an event count are negligible in comparison to the positive contributions.
%Even though
%event counts are in general complex, the imaginary portions
%of any macroscopic event count will likewise be negligible.
Furthermore, the erasure that a negative event performs on
the event count is total. It will not do to claim that the measuring device
can somehow remember the past, and the results from previous runs, for the device
has no memory.
%Suppose it were possible model
%some intelligent observer in the model.
It is true that
an observer, assuming he was in this case an intelligent being, could make
measurements on some memory devices that he has constructed, and thereby claim
to be observing past events. However, he can never
be certain that the history he thereby obtains is
consistent with what some external observer would
claim to be the past, (supposing that he could somehow
communicate with such an observer.)
Only by taking up the study of quantum mechanics
could the internal observer come to see at least the possibility that
negative events play a part in the physical world.
% although presumably, he might at first resort to a Copenhagen or many-worlds interpretation.
\vspace{\baselineskip}{\obeylines
\noindent{{\it Measurements}}\nopagebreak
\vspace{.6\baselineskip}\nopagebreak
}
%
The measurement procedure can be seen as the transformation
of physical configurations into language, or more precisely numbers.
Because the counter is always the means by which this transformation takes
place, on a fundamental level, the only permissible
statements that can be made
are of the form, ``Since the time this measurement run began,
this device has encountered $N$ events.''
This is even in the case of macroscopic measurements,
though in such a case, $N$
will of necessity be a large number whose value is unchanged
by adding or subtracting a few events. Consider a photomultipler tube
that has just discharged. The perception of there being
one single event that has just occured must not
confuse the fact that what we are observing is in fact the
state of the photomultiplier tube, and the location of the
large number of particles comprising that device.
In this interpretation, there is no dividing line between the microscopic world,
where particles appear to be wavelike, and the macroscopic world,
where objects are ``classical''. What separates the two is
that in the latter, one is choosing
integration paths involving huge numbers of events, so that
the law of large numbers comes into play, leaving only the
effect of the predominant positive events.
This is not to say that by using photomultipler tubes,
photoelectric devices and other quantum mechanical
devices, one is unable to observe the microscopic world.
Rather, what is being stressed is that the outputs
of those devices are macroscopic.
%There is a necessary conceptual leap in going from say
%the physics of what is happening in an electron beam
%experiment, to the macroscopic physics of the photographic plates, and
%other macroscopic measuring devices that are use to record
%the behavior of that beam.
%
Neither do the arguments presented here deny the existence of statements
that speak {\it about} the number of encountered
events. There are indeed other statements that can be made than
those of the form considered above. However, in making such me\-ta\-state\-ments,
one involves manifold and simultaneous interactions among regions of the
brain that are large with respect to
the atomic level. The statements involving counters
are important in that they provide a way of
turning physical phenomena into numbers.
%Though these fundamental statements we may make might
%seem crude or primitive, they serve here as the building blocks
%of perception.
Even though the
dynamics of the brain process that comprise
perception and consciousness are unknown, if they exist at all,
one can still have confidence that the microscopic
counting procedure will in fact lead in the suitable
limit into a regime where ``commonsense'' me\-ta\-state\-ments
are seen to be true. This comes from the realization
that macroscopic objects are large $N$ limits of
microscopic ones, and from the way that
the model, on a microscopic level, correctly describes microscopic events,
because it agrees with the formalism of quantum mechanics.
%does agree with the formalism of quantum mechanics, i.e., microscopic physics.
%
It must be stressed that this paper makes no
claims about the relationship of
these ``negative events'' to antimatter, or
to concepts such as, say, the virtual mass states
of quantum field theories.
It may indeed turn out that physically
isolating the negative events in a given distribution
from the positive events that outnumber them requires
energies so large that
particle-pair creation processes must be taken into account.
Likewise,
the conventional view of
a fermi sea, or the phenomenon of vacuum polarization, which implies a swarm
of companions within the
vicinity of a charged particle, may have relevance to
the fact that, in the model, even single particle states may
contain large numbers of lattice particles. For present purposes, however,
negative events should be seen merely as something analogous to
the negative regions of a Wigner function, so that they are
commonplace and
abundant even in low energy states.
%It may be pointed out that the probabilities
%considered in [TW] are all in fact positive,
%and that negative events occur only because of the manner
%in which some events are weighted. The reason for the persistent
%references to negative probabilities is that
%this weighting procedure is reflected in the measuring process,
%and it is only through the measuring
%process that one perceives one's surroundings.
%Also, as was shown in [TW], this procedure of
%weighting certain particles, allows one to
%obtain the probability densities associated
%with the absolute square of wavefunctions.
It might seem that one can invoke negative events and the relativity of
histories to justify any behavior whatsoever. If this invocation were the
totality of the model, it would indeed serve little purpose.
Yet because (1) the model allows one to obtain the probabilities
of quantum mechanics, locally and in a
three-di\-men\-sion\-al configuration space, the model is indeed useful,
and because (2) quantum mechanics,
in the macroscopic limit, yields the macroscopic world around us, one can feel
equally confident that the model provides a universe
in which internal (and macroscopic) observers find to be identical to our own.
\vspace{\baselineskip}{\obeylines
\noindent{{\large {\bf External Paradoxes}}}\nopagebreak
\vspace{.6\baselineskip}\nopagebreak
}
%
It is worthwhile to consider in turn some of
the objections that the previous discussion may elicit.
As has been mentioned before, the primary reason
negative events are unobservable at the macroscopic
level is that quantum mechanics itself predicts that
at the macroscopic level (i.e.~the Newtonian limit),
phenomena appear ``classical''.
The resoning is as follows: first, the distribution of events corresponding to nonpositive
probabilites
is always such that they are outweighed by the positive probability
events$^5$
%\footnote{
%The ``Negative Probability'' paper of Feynaman
%cited above, contains several elementary examples illustrating this point
%with respect to the Wigner function, for which the same can be said,
%and shows
%how negative probabilities can be used to generate
%the distributions of particles arising in such seemingly
%nonlocal phenomena as the EPRB experiments.}
(this is because the weighted sum of
the probabilities equals the absolute
square of the quantum mechanical amplitude for the physical process in question).
Therefore, in any large (macroscopic) arrangement of
events, the effect of the positive events predominates
that of the negative events. Now human beings are necessarily macroscopic
beings, and anything they perceive (even though it be the output of some
microscopic measuring device) is likewise macroscopic.
Therefore, human beings only perceive the world as being composed of
positive events.
To dismiss negative events as absurd is as valid as
dismissing the wave-particle duality, since this too,
cannot be directly perceived at the macroscopic level;
yet both concepts are useful in interpreting the physics
of the microscopic realm.
%It is to be expected that
%we who can only observe macroscopic things
%are unaware of phenomena that only manifest
%themselves at the microscopic level.
%Also, we cannot escape our universe in an
%attempt to view it externally.
%Obviously, it is not true that one spills a glass of milk, and
%then spills a negative glass of the same in order to clean up,
%and one would be mistaken to think our model permits
%a behavior that is clearly incompatible with the
%macroscopic, ``classical'' limit of quantum mechanics.
%As in the
%Wigner function formalism, the negative probability
%regions of phase space are
%inseparable from the larger positive probability regions, so that
%in the macroscopic limit the negative probabilites become
%unobservable.
Another important point
is that the measurement
process, as presented here, is not something that occurs once and
for all. For example, suppose some internal beings make a recording
of the clicks of a Geiger counter with
a tape recorder. Then every time they play
that recorder back, i.e.\ every time
they construct a picture, or history, of the
world around them, the measuring process begins
anew. Having done this, they cannot guarantee
that, to an external observer watching them evolve,
the history they believe in now is the same as what that external
observer would say they
believed in when they last perceived the Geiger counter record.
This is so, even if what they observe is a thing
as macroscopic as a section of magnetic tape, or a lab technician's notebook.
So long as the correlations provided by the model are
microscopically those predicted by quantum mechanics,
then macroscopically, quantities will correspond as well.
Even if some internal individual tried to take a photographic snapshot
of an event at some time, and then retake the snapshot
at some later time, and finally compare the two
photographs in order to try and notice
something like a negative event,
the final act of comparing supersedes
the previous perceiving of each individual photograph.
If, after comparing the two photographs,
the individual again looks at the first one, he again
supersedes the previous history he had
when he compared the two photographs.
If he next compares the first photograph with
his {\it memory\/} of how it
looked when he first saw it,
this is yet another measurement, and so on.
Thus, in order to
answer questions about
(i.e.\ make measurements on)
a particular measurement,
one has to step aside, so to speak, from
that original measurement. Only then can
one study the correlations among the original
set of measurements. As long as the correlations
that indicate some violation of causality
(e.g.\ a decrease in the net number of received
particles, indicating negative events) are given by
events that cancel in any
macroscopic limit, the internal
observer will have no reason to assume anything is amiss,
and will continue believing his commonsense notions of what is happening.
The distinction
between measuring something and then asking questions about
the measurement is a necessary one, and
makes possible the manifestation of the nonintuitive rules of quantum logic.
Consider also the noncommutativity of measurements
taken at different spacetime points (separated by timelike intervals). That is, in
``stepping outside''
of perceiving one phenomenon in order to perceive that
phenomenon's correlation with something else,
there is necessarily some traversal of spacetime involved.
Therefore, quantum mechanics, with its associated uncertainty principles,
its noncommutativity of separate measurements, etc., itself
preclude the ability to exactly identify perceptions that one had in the past
with the perceptions one has later.
Moreover,
there are a vast number of internal pictures of what
is happening at any given time, depending on the
integration paths used to obtain those pictures.
Two different observers may have radically different perceptions
of their surroundings. However, both choices yield
the correct microscopic results, and therefore, in
the macroscopic limit, the correct classical results. Therefore,
each of the observers will be convinced the universe he sees
corresponds to classical notions of causality.
It is true that many macroscopic events are such that
purely classical models would suffice to explain them.
In such cases, what one thinks one sees, what other nearby observers
think they see, and what
an external observer sees, are much the same.
Nevertheless, there are situations in which
%what we perceive as
radically different outcomes of some physical process
are actually closely connected in configuration space by
only a few microscopic events (e.g., a dead Schr\"odinger cat in a box versus
a live one; a disharging photomultiplier tube versus one at rest).
In such a situation, {\it both outcomes are contained
within the model.} The one that an internal observer believes to be the
actual outcome depends on the integration path he has chosen to observe the
distribution of matter. The internal observer is unaware of any paradox because
of each view's internal consistency with quantum mechanics, and its classical limits.
One can of course ascertain in a roundabout way these
nonintuitive aspects of the microscopic world.
The study of quantum mechanics has indeed done just that.
Aside from the uncertainty relations mentioned above,
one can add, e.g.,
the violation of Bell's inequalities, and the ability of Wigner functions
and other generalized probability distributions to readily explain
certain phenomena, at least in a formal way.
% not to mention zero-point fluctuations and fluctuations
%in the fermi sea;
All these are indications of the nonintuitive nature of
%
microscopic phenomena and of the need to extend
one's commonsense notions in order to be able to understand such
phenomena.
%
%manifestations that suggest
%that negative events may indeed be real and that it is
%quantum mechanics and the associated uncertainty
%principles themselves that stand in our way of directly observing them.
\vspace{\baselineskip}
\noindent{{\large {\bf Examples}}}
\vspace{.6\baselineskip}%
\noindent{{\it Macroscopic measurements}}
\vspace{.6\baselineskip}
Consider again the simulation of a universe in which a
double slit diffraction {\sl gedankenexperiment} is being performed,
as
presented in [TW].
%those who have not read this section in the previous paper would be
%well-advised to do so before reading the rest of this one.
The way one typically explains the workings of the recording
equipment, say a Geiger counter, used to detect the arrival
of particles at the observation screen, is to revert
to the ``particle'' mode of the wave-par\-ti\-cle duality.
That is, the explanation consists in saying that an incoming ion is
accelerated to high velocities by electric fileds within the
Geiger counter, whereupon it crashes into a metal plate producing
cascades of other electrons, and so forth.
One can, however, retain the wave picture
of the si\-tu\-a\-tion.$^6$
%\footnote{
%See for instance, N. F. Mott's explanation of
%$\alpha$-ray tracks from purely a wavelike perspective, in
%{\it Quantum Theory and Measurement\/}, J.~A.~Wheeler and W.~H.~Zurek (eds.),
%(Princeton University Press, Princeton, 1983).}
Because a beam of electron wave packets is impinging
on the observation window of the Geiger counter, the state
of this physical device at some time $t$ during the {\sl gedankenexperiment\/},
call it $\psi$, can be written
as a sum over many possible outcomes. Among them will be the state in which the counter
is silent, and the state in which the counter has just emitted a click.
In other words,
\begin{equation}
\psi= \left(\ldots+\alpha_q\psi_q +\alpha_c\psi_c+\ldots\right)\ ,
\end{equation}
where the $\alpha_q$ and $\alpha_c$ are the respective amplitudes for the
Geiger counter to be quiet in the moment under consideration,
and for it to have just emitted a click.
Note that each of the above outcome states is {\it complete}. That is, if there is
some intelligent being next to the Geiger counter, then $\psi_c$
is the state in which his neurons are so configured as to have
this being pondering and remembering the click that has just been
emitted. Likewise, $\psi_q$ would have him in a state of anticipation for
an upcoming click from the Geiger counter, that he knows may or may not come.
The point to be made is
that, in determining the dynamics of the beam, all one can do is
to make macroscopic measurements on the Geiger counter.
When one make a measurement on this Geiger counter, he sweeps
over large distances to determine the positions of the atoms and
molecules that comprise it. (Of course, the internal observers
of this experiment would actually make measurements on the regions of
their retinas and eardrums that are oriented toward the apparatus, but the result is the same.)
Having made these macroscopic measurements,
the observer will find the results to be consistent with those of a quiet Geiger counter,
or with those of one just having emitted a click.
%
%
%Given
%that this device is huge with respect to the macroscopic scale,
%and that quantum mechanics predicts that for large numbers the
%event counts we obtain are positive, one cannot claim that
%on the microscopic scale there are no nonpositive events any more
%than one can deny that there are wave-like phenomena at those microscopic scales.
%
%
It may even happen that an external observer
making measurements, depending on the specific integration path
he chooses to collect the data, may find one integration path
is consistent with $\psi_c$, the other with $\psi_q$.
Because
both of these paths involve complete and classical situations,
the internal observers in each of them would be unaware of
this startling duplicity.
%The wave-like phenomena qua negative events, (not to mention the noise
%and experimental uncertainties that
%always must be taken into account,)
%preclude our retroactively matching a given, macroscopic
%click on the Geiger counter to a unique electron.
If this explanation seems to lean too much on the
wave-pic\-ture, how is it possible, one may ask, to arrive at the
conviction that quantum phenomena are discrete?
The answer to this lies in the fact that discreteness is a property determined by correlation
experiments. A coincidence counter, set up to locate two
separate and simultaneous outputs from a dilute stream
of single particle states, does not register any events, when
suitable corrections for experimental and other errors is made.
Such are the experimental manifestations of discreteness.
Moreover, something similar to this coincidence experiment occurs even in our brains,
and is what allows us to perceive the discreteness
that characterizes a pulse, a click, a single track, etc..
And as was shown in [TW], the model presented here is
able to obtain correct multi-particle correlations as well.
%
%
%BE SURE TO PUT IN SECTION SAYING THAT EVERYTING HAPPENS IN THE WAVE PICTURE.
%THE WAY WE GET THE DISCRETENESS OF EVENTS IS BY DOING CORRELATION MEASUREMENTS,
%EG, BY SEING THAT THE EVENT IN WHICH A SINGLE PARTICLE IS AT TWO
%PLACES AT ONCE VANISHES.
%
%
%
It is interesting to recall that Born$^7$
%\footnote{
%M.~Born, in
%{\it Nobel Lectures in Physics\/} (Elsevier Publishing Company,
%Amsterdam, 1936), p.~256.}
proposed his interpretation of the square of the wavefunction
as a probability density primarily as a reaction against
Schr\"odinger and those in the wave-mechanics camp. He saw this
formalism as trying to undermine the essentially discrete nature
of microscopic phenomena. The formalism of generalized probabilities
developed here allows an alternate method of resolving this dilemma, and of representing wave-like phenomena by
discrete particles.
It is true that the detail of the exact workings of the Geiger counter, and
the quantum effects of eardrums and so forth that allow one to discern
the counter's operation are not fully known. But as stated before,
the model can be expected to yield the correct macroscopic results because
it yields the microscopic results of the quantum formalism, and
because of the success this formalism has had in explaining physical phenomena.
\vspace{\baselineskip}{\obeylines
\noindent{{\it The EPRB experiment}}\nopagebreak
\vspace{.6\baselineskip}\nopagebreak
}
%
A useful means of illustrating some of the paradoxes
involved in the multiplicity of integration paths is provided
by considering a simulation of the EPRB {\sl gedankenexperiment}.
An idealized form of the Einstein, Podolsky, and Rosen experiment,
first proposed by Bohm,$^8$
%\footnote{
%For example, Chapter III, Wheeler and Zurek, cited above.}
consists of a laboratory where
singlet states of two electrons are created. These states
are unstable, and eventually they explode, sending their
two constituents in opposite directions; the reaction
Hamiltonian is independent of spin. Far away
to either side of the singlet creation site are
two electron polarizers, or Stern-Gerlach analyzers (Fig. \ref{fi:eprb}),
denoted as A and B.
%
\begin{figure}
\psfig{file=hx/twfig/twf-eprb-2.ps,height=1.7in,width=3.0in}
\vspace{1.0\baselineskip}
%box is 7 1/4" by 3 7/8", lower left corner 1/8" 6 7/8"
% upper right corner 7 3/8" 10 3/4"
%Have a checkered lattice of points connected
%by dashed lines.
\caption{The EPRB experiment.}
\label{fi:eprb}
\end{figure}
%
An electron that passes through the observation
window of a polarizers is sent to one of
two output channels, the ``up'' and ``down'', depending on whether
the spin is measured to be $+{1\over2}$ or $-{1\over2}$
with respect to the axis of the polarizer, which lies
in the plane of the polarizer's observation window.
It will na\"\i vely be assumed that it is possible
to construct an accurate Stern-Gerlach analyzers for electrons, since doing
otherwise would needlessly complicate the presentation. Also, if one is
uncomfortable about using spin as a dynamical variable, given that
spin has not been discussed in terms of this model, one can rephrase
the experiment with spinless particles in terms of angular momentum, linear momentum, or
even in terms of two sets of double slits situated on opposite
sides of the particle sources.
Recall from [TW], that a two-particle event may be of one sign,
even though its constituent one-particle events, considered alone,
are each of a different sign. Assume now that there
is some being standing by one of the polarizers, writing
down the series of incoming up and down electrons along with
their times. Consider the possible
outcomes for this experiment at some specific time, just as in the above example
concerning the Geiger counter.
Each of these outcomes would have this being sitting by the polarizer, with
the lab notebook before him containing a faithful account
of the corresponding outcome state, say a list of pluses and minuses.
Each of the outcome states
extends to all aspects of this individual, so that if we,
being external to the model, were
able to ask him about what he
has just seen, even his brain cells would be so configured
so that he could give us an accurate description.
Suppose there is another such being at the opposite polarizer,
and that after a certain time, they get together in some location
and discuss their results. Now the microscopic events that
made up the experiment have different phases when
considered as multiple events. Indeed, as shown in [TW], it is even possible
for there to be a multi-particle event without having
composite single-particle events, and vice versa.
Nevertheless, the results at
each of the two polarizers when considered together, i.e.~when
put through coincidence counters and other correlation
apparatuses (such as the brains of these two individuals), are such as to be in
accord with quantum mechanics.
This feature of the model extends even to
the macroscopic regime,
to the two observers.
% will eventually be nodding to each other
%in agreement after checking over their results.
That is, for the given relative orientation of
the polarizer's axes, the clicks at one polarizer
will be confirmed to be coincident with the clicks at the other.
Also, the resultant rates of ``$+ +$'', ``$+ -$'', ``$- +$'' and ``$- -$'' events
(where ``$+ -$''
indicates a coincidence of a spin ``up'' electron on the first polarizer and
a spin ``down'' in the second one, etc.) will be in accordance with the predicitions of quantum mechanics,
and therefore inviolation of Bell's equality.
Such is the case even though external observers, looking on the simulation,
could see
that what the two notebooks are now recording as having happened in the
past is not the same as what the notebooks were recording initially.
But the memories of the two beings
are such that their perceptions of what has happened is, apart
from the violation of Bell's equalities, in accordance with commonsense
notions of causality, and the ``absolute'' nature of the past. Were the
external observers able to communicate to the internal ones
what was ``really'' happening, the two internal observers would
probably find the communication absurd.
This multiplicity of the descriptions of the past occurs because the measurement processes
at the two individual polarizers, and then at the common destination,
involve different integration paths, as well as a difference
of types of measurements (i.e., single vs. mul\-ti-par\-ti\-cle events).
It is assumed in this discussion that the coincidence
counters and the neural mechanisms the two internal
observers employ perform
in some way the multi-particle event counting
as described in [TW].
Suppose that later on, one of the internal beings looked at his notebooks while alone again.
Depending on the integration path he uses to observe his notebook,
it is possible that he might (unconsciously) go back to his original
picture of the experiment, i.e.\ before he conferred with his partner.
Moreover,
%he would tell us, if we could ask him, that he remembers having
%met with his lab partner and seeing that the
%events he sees now (and what we say he originally saw)
%are such that when considered together with his partners,
%the entire set of events would be consistent with quantum mechanics.
suppose he could reference (i.e.\ make measurements on) his memory cells,
in order to tell some external being what he {\it remembers} seeing in his lab partner's
notebook.
%he might tell us, and
The external being could see that
he has in fact unknowingly changed some of his partner's results, but again in such
a way as to make the respective rates of the ``$+ +$'' and other events
consistent with
what quantum mechanics says they should be.
Again, the reason for believing that this will be the
case is the agreement of the model with the
quantum formalism, along with the conviction
that quantum mechanics will, in the appropriate macroscopic limits,
provide an accurate description even of classical phenomena.
\vspace{\baselineskip}{\obeylines
\noindent{{\it Classical objects}}\nopagebreak
\vspace{.6\baselineskip}\nopagebreak
}
%
Consider next the notion of paradoxes involving
classical objects.
A classical object is one whose position and momentum can be specified with arbitrary
precision.
The
perfectly reflecting or absorbing walls of a box, the potential
barrier at which particles are fired, such
are the classical objects one typically employs.
Note that these objects are not necessarily `fixed' in the discrete
lattice that is configuration space of the model. They may instead
translate in such a way that, when considered over macroscopic time scales,
the paths of the objects converge to their desired
continuum limits.
Even when classical objects are merely mathematical artifices, whose
very presence is inconsistent with the belief that {\it all}
objects must obey the laws of quanatum mechanics, the convenience they provide
makes one loath discarding their use altogether.
%In many cases, such objects are mere conveniences; if
%one beleives that the world is totally quantum-mechanical at
%all levels, then one believes that there are no such classical elements.
As was mentioned previously, the incorporation
of classical events can be especially problematic
when
perceptions of the past are nonabsolute. For example,
consider the simulation of some physical being or device
that has initially been programmed to
``do X if Y occurs.'' It is conceivable
that the outcome of this simulation is
one that is perfectly consistent with quantum
mechanics, but is such that the events
have been erased in such a way that the
being is now programmed with the instructions
``do X if Y does {\it not\/}
occur''. Therefore, in such a situation, the program itself must be
interred into the simulation as a classical object.
Similar difficulties are encountered in the
simulation of classical instruments such as photomultiplier tubes
which experience a
macroscopic ``dead time'' after discharging.
In general, it is not difficult to include classical objects
if one is content to
simulate phenomena in which the classical
objects remain always unchanged by what
is happening, e.g. situations involving perfectly reflecting barriers.
(Of course there still may be other constraints, such as relativistic ones,
that such classical objects may be unable to satisfy.)
If, however, the state of a classical object
depends on the particular outcome of some certain event,
then the classical object must itself be described by a superposition of states,
so that it ceases to be a classical object.
%,becoming therefore like a Schr\"odinger cat.
There are several ways to remedy this situation, given that
external beings have powers beyond those
of internal ones. For example, one can
have classical objects judiciously emit particles
in such a way that the resultant negative event count increments
``erase'' other events, leaving the desired outcomes.
Of course doing this runs the risk of simulating outcomes that
are highly improbable. However, if one is interested only in seeing
what would happen, given a certain occurence, without regard to the
improbability of that occurence, such an approach may be useful.
%
%For instance, in a Schr\"odinger cat
%experiment, we may {\it ad libidum\/} emit negative gamma rays from
%some point in the space in such a way as to cancel any events that might lead
%to the death of the cat. Since quantum mechanics is itself a probabilistic theory,
%the situation in which the cat lives a very long time is always possible,
%but depending on the circumstances may be highly improbable. The problem
%is thus related to the question of how much nonrandom manipulation of
%a random sequence is possible before the sequence becomes effectively
%nonrandom.
%It is of course, possible to do away with classical
%objects , and indeed the notion of beings that
%decide on what to do in a way that is somehow independent
%of the physical laws which they must obey (i.e. in such a way
%as to be external agents effecting internal changes on their surroundings)
%is out
%of place in a universe where everything obeys quantum mechanics.
%KILL THIS PARAGRAPH
%Note also
%that unless we added some extra constraints,
%the total number of fermions in our universe would, externally
%speaking, appear to fluctuate, but macroscopically,
%the law of large numbers would make such effects
%negligible.
%External observers of course, would still need to go through
%the protocol of picking a volume and integrating the
%events within in order to obtain numerical results
%of what is happening,
%%, and again, we would see that each
%%selection of integration paths might result in a
%%distinct picture of what is happening within the universe
%%we are simulating.
%and eventually one must confront how it is that internal
%observers come to perceive a picture of what is going on, but this
%is outside of the scope of the present inquiry.
%EVENTUALLY I WILL KILL THIS PARAGRAPH, BUT I SHALL KEEP IT IN ALL THE
%DRAFT NOTES.
%The problem of classical objects may apply to cosmological questions of the universe itself
%(ie, the ultimate boundary conditions). This can be carried
%even to questions concerning the nature of the model itself, if
%we dare to risk being inconsistent. For example, the formalism outlined
%above started with discrete events, that taken together by way
%of interference and cancellation, yielded systems with quantized
%energies, momenta, etc. One might ask if the discreteness
%of the particles themselves is similarly the net result of
%some more primitive subphenomena; the answer of how and why particles have the
%same charges and masses might perhaps be answered by such arguments, so that
%the notion of negative events, and relative histories can be useful
%in ways other than the ones shown here.
%%An example of generalizing concepts so
%%that they interfere is to make our
%%universe one pure wave. For one particle
%%that's easy, you just integrate the amount
%%of wave stuff in a region. Particle
%%quantization, or the fact that things come
%%in pulses would then be another form of
%%quantization, in the sense that any
%%experiment designed to test the density
%%of information, would yield events that
%%interfere to zero except in the neighborhood
%%of integer values. Although
%%conceptually much more natural,
%%the process {\it at the present\/} seems
%%unwieldly for many particle systems
%%(how does one construct a wave in some
%%3 x n dimensional or perhaps even 3 x
%%dimensional waves) and so is not
%%attempted at the present.
%However, it may prove useful to invoke
%this type of argument to explain
%why particles come in quantized charges,
%masses, etc.
%finding the boundary cond
%In closing this section, it is important to note that we
%do not claim that the universe is
%simply the evolution of some lattice wave
%of bra and ket particles,
%any more than one would claim that all
%diffusion phenomena consists of particles
%meandering on a grid. ????The lattice
%waves and paths on a grid are clearly idealizations,
%that are open to further realizations.????? Even so, just as
%particle paths on grids can in a sense
%explain diffusion, the lattice waves,
%in conjunction with the interpretation proposed here,
%can in an analogous way explain quantum mechanics.
%\section{Lattice waves}
%THIS PARAGRAPH HAS BEEN TRANSCRIBED ABOVE.
%For the moment, we make no restrictions on how the counter sweeps across the
%regions of space in which the measurements are being made. When the
%phenomena we simulate can just as well be described classically, the
%integration path will not matter. However, in certain situations,
%the choice of the path radically changes the resulting picture of
%what is happening. We will discuss this in more detail below. For
%now, note that just as the choice ofGaussian surfaces is
%arbitrary, and does not affect the dynamics of the physical event under
%consideration, {\it any\/} integration path will yield a picture
%consistent with quantum mechanics. That is, the beings within
%the picture have no reason to suppose that quantum mechanics is
%being violated. Therefore, since there are obviously a large
%number of possible integration paths, we are in effect {\it describing
%a large number of universes with each simulation\/}
%One might immediately object that this is not what
%our experience tells us happens. In a region
%of cancellation, even if we suppose that
%negative events are unobservable in themselves, we do not register
%a certain number of events and then notice later
%that the number has decreased.
%To this we reply that we must view this situation
%{\it internally\/}, i.e., as an observer within
%the universe would see it.
%Note that
%the click of a Geiger counter, or
%the speck of precipitation on
%a photographic film we usually associate
%with a the arrival of a particle diffraction experiment
%is not itself
%the direct result of a particle encounter.
%These are complicated macroscopic events, involving
%large cascades of electrons, or precipitate molecules,
%and our perceptions are based on the macroscopic state of these collective phenomena.
%The corresponding counters therefore encounter and compare macroscopically
%large numbers of events, so it is not surprising that we do not notice phenomena
%whose effects are in comparison wanishingly small.
%Nevertheless, our model is still useful, for the {\it numbers\/}
%that it gives us are correct; the net number of events is indistinguishable
%from what quantum mechanics allows, for quantum mechanics itself
%is a probabilistic theory that can be verified or disproven by
%recourse to a large number of measurements.
%Even so, the dependence of the
%outcome of measurements on
%how the measurement takes place can be made to affect even macroscopic events.
%To illustrate this, imagine simulating in our model some macroscopic device that say, rang
%a macroscopic bell (or in Schr\"odinger's more familiar version, cracked an ampule of
%poison gas, thus killing a cat) when the Geiger counter to which it is connected registers
%an event. We assume all these devices are imbedded within the model universe.
%Now the way in which external observers get a picture of what is going on is
%by sweeping (at every instant of time) across every point of the universe with
%their counters, much as a submarine radar sweeps its space in order to reveal
%information about the surroundings.
%For the sake of convenience, assume for the moment that our universe is one-di\-men\-sion\-al
%and of finite length;
%by sweeping the counter from one end of the space to the other, we obtain at each point of
%the space the quantity
%$${\cal C}(x)=\int_{x' < x}N \,dx'$$
%It is through this cumulative number of events we can get a picture of what is happening in the universe.
%It is true that
%unlike the abovementioned submarine radar, our events can be positive or negative, but where macroscopic
%numbers of particles and integration volumes are involved, the above function is always increasing, thereby
%yielding a sensible classical picture. Where the presence of negative events may have a
%macroscopic effect is in the region outside the Geiger device, where depending on how we
%sweep the space, a negative event may cancel a previously registered, positive one so as to yield a situation
%in which the bell is yet unrung. If an alternate sweeping path (say, integrating along the negative $x$-axis)
%made the negative particle change the number of particles within the region comprising
%the Geiger device instead of the region outside its measurement window, then this path might
%yield a situation in which the bell was already rung.
%The two situations are
%so closely connected in the configuration space of our model universe that the
%adjustments made by the registering of a negative event are sufficient to turn one event into the
%other. This is perhaps the ultimate paradox of quantum mechanics. However, if we were to simulate
%internal beings in our universe who are also watching the
%outcome of these events, they could not be seen by an external observer to appreciate
%their paradoxical plight,
%for regardless of the way an external observer sees such beings, he sees them in a world
%where the large-scale phenomena appear to obey the laws of classical mechanics, and
%therefore the laws of common sense; they themselves would think the bell had rung, or that
%it had not, and they might be shutting their ears to drown out the noise or might not, and so forth,
%but they would find it absurd to think that the two scenarios could exist simultaneously, so to speak.
%%It is true
%%that in any case, even where we might expect total cancellation of
%%amplitudes, there might be some events registered, but as long
%%as the fractions involved are close to zero, the effect may be
%%attributed to noise, or the uncertainty of the initial conditions, etc.
%The dynamics of the bra and ket particles of our model
%is only slightly removed from that of a random walk coupled
%with a branching process\footnote{Hrgov\v{c}i\'c, {\it op. cit.}}.
%An observer looking in on a simulation would (on a microscopic level)
%see no long-range order in the behavior of any one particle;
%in other words, what we thus have is an {\it ensemble}
%interpretation. In simulating the wave packets corresponding to particles
%possessing a
%relativley well-defined momentum, as in the dif\-frac\-tion-slit
%experiment above, it is only collectively that
%the corresponding bra or ket particles describe
%the large-scale structure of the resultant wavepackets.
%The individual particles themselves do not have a
%well-defined momentum, or indeed any property outside of their
%initial position and phase; it is only an ensemble of such particles
%that can be said to contain such information.
%
%If instead of passing the above wave packets through
%slits, we were to pass it through a diffraction
%grating, and set our counters some distance away,
%we would find that only in one region does the
%counter have a nonvanishing collection rate, the
%spread and location of this region depending on the
%length and wavenumber of the ensemble of particles.
%Furthermore, the rate of the events collected
%in this one region are the same as the net rate of
%all the events collected on the second screen
%in the diffraction grating. That is, we would
%be unable to detect a loss of particles in comparing
%the two experiments.
%We can extend our results to simulate multi-particle
%phenomena as well.
%To familiarize ourselves with the procedure involved,
%let us imagine again the very first simulation we
%considered, that of a large number $\cal N$ of
%baxes. Suppose instead that we have $\cal N$ {\it pairs\/}
%of such boxes, with the member of each pair being denoted by $A$ or $B$.
%Each of the two sets of boxes has a unique coordinate system associated with it.
%Also, we suppose for the present that each pair of boxes is sufficiently separated so that
%interactions between them are negligible.
%Just as in the first situation, where we assigned a distribution of
%particles such that by means of probing the
%boxes at analogous points and times, we were able to
%obtain (supposing that $\cal N$ was large) the wavefunction
%$\psi({\bf x}_p,t_p)$ at the corresponding points ${\bf x}_p$ and $t_p$,
%we now assume that the distribution of particles
%in the $A$ boxes is such that if we were to repeat this procedure
%we could determine some wavefunction $\psi_A({\bf x}_p,t_p)$
%and likewise that the distribution in the $B$ boxes corresponds
%to some wavefunction $\psi_B({\bf x}_p,t_p)$.
%Then we can consider two sub-counters, $A$ and $B$, respectively
%probing the $A$ boxes at the points corresponding to some point and time ${\bf x}_1$ and $t$,
%and the $B$ boxes at the points given by ${\bf x}_2$ and $t$.
%The counter increment corresponding to the composite event of finding a particle in box $A$ at ${\bf x}_1$ and $t$,
%and a particle in box $B$ at ${\bf x}_2$ and $t$, is then only nonzero if a bra and ket particle
%are simultaneously found at the place in question in $A$, and a bra and ket particle are
%at the same time simultaneously found in the point in question of the box $B$ that is the
%partner of $A$. The sign of the increment, given that the bra and ket particles are
%positive, negative, etc., is obtained by multiplying the signs corresponding to
%the ket particles times the complex conjugate of the signs corresponding to the
%ket particles (the sign corresponding to positive, negative, positive,
%po\-si\-tive-ima\-gi\-na\-ry, and ne\-ga\-tive-ima\-gi\-na\-ry particles are of course
%respectively, $1$, $-1$, $+i$, and $-i$, just as in the one particle protocol above).
%Indeed, note that we can define the interactions between particles
%so that the above considerations are reducible to the
%protocol for the single particle events. For example
%we can imagine explicitly constructing a physical measuring device internal
%to our model containing
%an {\sc and} gate that is designed to emit a particle every time two
%events happen in a certain way, along with two probes, one leading from an $A$ box
%to the and gate, the other leading from the $B$ box to the {\sc and} gate. Then by setting our
%conceptual counter in the output region of this measuruing device,
%we may reduce the measurement of multi-particle phenomena to the measurement of
%single-particle ones. Thus, the formalism for counting multi-particle events
%that we have outlined in [TW] can be subsumed into single-event measurements,
%provided that the physical interactions of the {\sc and} gate, as yet unspecified,
%yield events with the same phases.
%Bogus, but interesting, paragraph.
%ACTUALLY COUNTING CAN EVEN BE SIMPLER
%THAN THAT. CAN TRANSFORM EVEN THE COUNTING INTO A SERIES OF YES-NO QUESTIONS,
%SPECFICALLY, HAVE THE NUMBER OF EVENTS EXCEEDED SOME THRESHHOLD, AND
%TURN THE REST OF THE EVETS INTO SERIES OF or AND and QUESTIONS.
%Suppose the probe encounters in box $A$
%a po\-si\-tive-ima\-gi\-na\-ry ket particle and a positive
%negative bra particle. By the protocol given for the single particle case, such an
%event would not increment the counter if we were
%just observing what goes on in box $A$, for the sign of the event
%would be imaginary. However, if in box $B$ we simultaneously found a bra and ket
%particle of the same signs, then the counter measuring the composite event
%increments a counter that is measuring the multi-particle event by $+1$. That is, the two masurements
%described are different; mathematically, this is indicated by the fact that
%the associated quantum mechanical operators do not commute.
%In the real world, as well, the perception of some
%single event, and the perception of the {\it comparison\/} of that event with some other one
%are two different processes, coresponding to different recording devices. Yet because
%each way of looking at the event yields a result that is consistent with quantum mechanics,
%and therefore macroscopically with classical mechanics, the situation appears to all internal
%observers as our universe appears to us.
%Generalizing further, we can model arbitrary multiple particle
%events (see Appendix).
%%%%%%%%%%%%%It should be noted that
%if we are to use the random walk dynamics proposed, which
%calls for the particles involved to replicate at every
%discrete interval of time, then it is vital that
%the random initial particle locations are assigned so that the
%particles obey Fermi-Dirac statistics. We shall see how this
%%%%%%%%%%%%%%%is done in the Appendix.
%{\it A priori\/}, there is nothing in
%the model we propose that forces the simulated particles to obey
%Fermi-Dirac or Bose-Ein\-stein statistics, although
%in constructing a lattice on which we might simulate
%physics, the Pauli Exclusion Principle, in which limits
%the number of particles that may be at a given location, seems
%to be the most practical. In fact, as the Appendix will show,
%fermionic behavior can be readily incorporated into
%our lattice wave dynamics, and of course, bosonic particles can be built up as composite
%structures of fermionic particles, so that one
%can simulate both kinds of statistics.
%Because the random walk dynamics of
%the particle obeying a given wavefunction is coupled to a branching process,
%in principle, one could observe a particle interacting with itself, so that
%even a one-particle event might yileld a situation in which there is
%at any given time, more than one event ocurring; i.e., a situation
%suggesting that a particle is in two places at once. However, we
%can correlate the offspring particles that result from
%any one-particle branching process in such a way as to make
%the {\it net\/} result of any two-particle measurement on
%a series of one-particle states to be zero, so that on the
%macroscopic level such phenomena can be said to not occur.
%Multi-particle observations on differing particles, i.e. particles
%that do not have a common ancestor in the branching process and
%are therefore uncorrelated, can be made to yield amplitudes for
%multi-particle events that are given simply by various combinations
%of products of the single-particle amplitudes. We give an
%example of how this can be done in the Appendix.
%The paradoxes of the EPR experiment are other manifestations of
%what we have already encountered, namely the necessity of recognizing
%the distinction between observing a multi-particle event and
%observing one of its constituent one-particle events. Suppose an experiment consists
%of two Stern-Gerlach devices (S-G) symmetrically situated between a
%region in which a spin-0 particle explodes into its two spin-$1\over2$ constituents.
%The experiment consists of rotating the S-G with respect to one another
%and observing the correlations among the particles as a function
%of the relative angle between the two S-G.
%We may imagine placing a counter at one of the S-G, or the other, or
%at the output end of some {\sc and} gate which receives signals from each of
%the arms and increments according to whether or not there has been, say,
%a spin-up particle (i.e., a particle whose spins are measured to be along
%the same axis as that of the respective S-G) at each of the S-G, or a
%spin-up particle encountered in the first S-G with a spin-down particle
%encountered in the second, and so on.
%
%Note that we do not make
%any restrictions on the magnitude of the signals leading from each S-G to the {\sc and} gate;
%in a real-world experiment, they could be the decidedly macroscopic
%outputs of a photomultiplier tube; we only insist that the signals themselves
%must obey the laws of quantum mechanics.
%Alternatively, instead of explicitly
%simulating such an {\sc and} gate, we may simply carry out this procedure conceptually,
%to thereby see what the results would be if we did actually construct a device.
%As above, the the averages we obtain at each arm (i.e., the
%relative fraction of spin-up and spin-down events) are the same. However,
%from an external viewpoint, the particular histories held by two observers
%at each of the two S-G is different.
%
%For example, one may register
%a positive spin-up event at the same time the other partner receives
%a negative spin-up event, so that each of their counters (and pictures of
%what is happening) is altered accordingly. If they in turn, send signals
%to some distant third observer, then the signals, being of necessity
%quantum phenomena, must also be measured in terms of some counter, and
%the picture the third observer has is in turn different from that
%of the two informants. Even if all three observers were to quantum mechanically write down
%his results in their quantum mechanical notebooks, and then the third observer were to
%walk over to one of the S-G and compare notes, the correlation experiments they
%perform on the two sets of notebooks, would yield a still different history, but
%in all cases, a history consitent with
%their sensibility. That is, because the procedure above correctly gives quantum
%mechanical numbers, any events signalling some discrepency that cannot be
%accounted in terms of noise or experimental error cancel away.
%Therefore,
%they might claim that they have no direct evidence of the relativity of
%histories, or of quantum events; only an external observer would be able to
%see that their comparison of notebooks yielded by the comparison of notebooks yields
%a picture that is distinct (on the microscopic level) from all of the three previous ones.
%But also note that given the location where a counter is placed, there is no
%ambiguity or probabilism in how the counting is to be performed; the procedure itself is deterministic, and
%with the use of a suitable pseudo-random number generator which we initially use to assign particles
%in accord with the boundary condition, and thereafter use to generate the individual steps of
%the random walk processes, the entire evolution may be made deterministic.
%
%
%
%COMBINE THE NEXT TWO SECTIONS
\section{Problems of the conventional interpretations}
It is useful to consider other interpretations
that have been proposed as alternatives to the
Copenhagen interpretation, and their similarities
to the present one.$^9$
%\footnote{
%D. Bohm, {\it Quantum Theory\/}
%(Prentice-Hall, Englewood Cliffs, 1951), pp.~611-23.
%Some of the work cited h%ere, as well
%as an elaboration of the terminology and
%concepts within this section can also be found
%in Wheeler and Zurek, cited above.}
Although the Copenhagen interpretation is
the best known, it represents great difficulties in
providing a model that
reconciles locality with the correlations
found in, say, the EPR-type experiments performed
by Aspect, Grangier and Roger.
Even so, the Copenhagen interpretation has its own
kind of simplicity, in its stark division
between the numerous potentialites of the
wave-like microscopic realm, and the solid and certain actuality of
the macroscopic.
In the interpretation presented here, it is the microscopic
configuration of the universe that is
absolute, and, given that the derived macroscopic
pictures of the present and past varies from observer to observer,
it is the latter that exhibit multiplicity and diversity.
%???Note that there is no ``reduction of the wave packet''
%in our theory. Although in general,
%the perturbations incuded by a measuring device will
%indeed destroy the evidence of wavelike behavior,
%as we have mentioned before, the measuring process, i.e.,
%the transition from a microscopic event to a macroscopic one
%is not something that occurs once and for all. When
%we study the results of a previous measurement we have somewhere recorded,
%we initiate a process that is procedurally identical
%to that previous measurement.
%%, and externally speaking,
%%the picture implied by the original measurement,
%%and the review of that measurement need not be identical.?????
Some other proposed mechanisms and
`explanations' of quantum phenomena,
%given by E.~Nelson and others,
involve
generalizing the diffusion equation so that it
contains an imaginary diffusion coefficient,
thereby turning the diffusion equation
into Schr\"odinger's equation. There are other examples as well, in
this category of generalizing properties outside their usual domains of validity.
Reference has already made, here and in [TW], to the Wigner function formalism
and other theories involving generalized probabilities, as
well as computations of quantum propagators that
involve extending time to the imaginary axis.
Although these approaches formally provide correct results,
they do not by themselves indicate what
the generalized quantities such as imaginary diffusion constants,
or evolution in imaginary time might mean. To put it another way, they
do not indicate how such notions
can be applied to the construction of a three-di\-men\-sion\-al model of
quantum phenomena. Also, in the case
of the Wigner functions, one typically has to first obtain
the usual wave function and then subject it to an elaborate
transformation in order to arrive at the probability
distribution. Even in the case of the simpler generalizations of the
Wigner function, the evolution equations of these distributions
are unwieldly, and they are difficult to extend to relativistic
phenomena. Nevertheless, the present model even in avoiding these
difficulties has obvious
similarities to
the diffusion equation generalization, in that wave phenomena
result from particles whose dynamics is similar to that of the random walk.
Likewise, the model is obviously akin to the generalized probability theories.
%One may ask why we have followed more closely the Wigner function approach,
%adapting the negative regions into positive regions of
%negative particles, and so forth. The answer lies in the fact
%in order to get the Wigner function one typically has to get
%the wave function first; the transformation one then imposes
%on the wavefunction in order to arrive at the Wigner function is
%difficult to phrase in terms of a physical model, even though
%the Wigner distribution can be shown to uniquely satisfy
%some very simple constraints (that are albeit problematic, from a relativistic point of view.)
The ``many-worlds'' interpretation$^{10}$
%\footnote{
%H. Everett III,
%``\,``Relative State'' Formulation of Quantum Mechanics'',
%{\it Reviews of Modern Physics\/}, {\bf 29}, 454-62 (1957).}
is likewise
one that seems impossible to implement in a three-di\-men\-sion\-al
configuration space, unless some method can be found of
simulating all the ``branching states'' simultaneously.
There is again, however, a
similarity to the present interpretation, in that different
choices of integration paths can yield many different
pictures of what is happening in a simulation.
Since each integration path yields a picture that
is a valid one (to the internal observer making the
measurements), the present model can be said
to yield many (though not all) possible universes at once.
The pilot wave interpretations of quantum mechanics,
proposed by E. Madelung, L. de~Broglie and others,$^{11}$
has long seemed the most obvious approach for those who wished
to restore a classical framework to
quantum mechanics. Although such interpretations
present a much more tangible, mechanistic view of
quantum phenomena, and are threfore prime
candidates for a theory that can be implemented
in a three-dimensional configuration space, they are likewise
difficult to reconcile with EPR-type paradoxes. Bohm$^{12}$ does
give a model that is similar in some respects to a pilot wave theory,
and that does allow for such seemingly nonlocal behavior,
by making the dynamics local in {\it phase space.} However,
such a model cannot be implemented with interactions
that are local in a three-di\-men\-sion\-al configuration space. Even so,
these models are similar to the present one in
that they do ascribe macroscopic phenomena
to particles that
propagate
from point to point in a purely local fashion, which cannot be said of the
Copenhagen interpretation.
Stochastic electrodynamics is another theory
that has had some success in obtaining
quantum mechanical results through purely classical means.
By postulating a random distribution of electromagnetic
fields, T. Boyer has managed to obtain an energy
distribution within a black body that is consistent
with quantum mechanics and yet uses only classical electrodynamics.
Nevertheless, difficulties remain. For example, there is the problem of explaining how the
random background electromagnetic fields yield the long-range correlations
of the EPR experiments.
%Again, random motions, induced by the background
Still other semiclassical mechanisms
for such correlations have also been proposed. These
use the leeway granted by present
experimental uncertainties and
by so called ``loopholes''$^{13}$
%\footnote{
%D.~M.~Greenberger (ed.),
%{\it New Techniques and Ideas in Quantum Measurement Theory\/}
%Vol. 480, Annals of the New York Academy of Sciences
%New York, (1986), and references within.}
in order
to make {\it ad hoc} model that
has not yet been disproven, but so far, none has found
wide acceptance.
One way of explaining the EPR and other such correlations through
classical fields such as are used in stochastic electrodynamics brings up
an interesting feature of the present model.
Presumably, one may account for the EPR-type correlations by invoking
an total determinism. Note that all the particles
and fields of stochastic electrodynamics are
determined by the initial conditions, including
those comprising the experimental arrangements and
indeed the experimenters. To advocates of this total determinism,
the long range correlations one sees as the result
of some EPR experiment are
merely the result of some local correlations in the
initial conditions of the universe. (It should be pointed out that
as far as this author knows,
Boyer himself has never advocated such a claim, and one
can be a proponent of stochastic
electrodynamics without doing so.)
If one does adopt the view of total determinism,
there remains the finding of an explicit mechanism by which the correlations of
the initial state of say, an EPR experiment are reasserted through the
outcomes.
Even if one could be found, there remains a more serious problem in all models that
make the observed order contingent on
the predetermined behavior of the observer; namely, the loss of
free will. As with the notion of locality, free will is more
than just a philosophical conceit. It is also a
property that is enormously useful, even necessary.
One cannot hope to simulate systems interacting
with arbitrarily varying external fields and forces, perhaps subject
to the outcome of events which
they produce, and which are thus initially unknown,
if everything in the future and past must already be contained in
the boundary conditions.
To repeat what has been said before in another context,
one cannot expect that the nature of things is such that
everything can be simulated just as desired. It may ultimately
happen that the only sensible physical theory is indeed
one which absolutely determines all human actions.
%, even the action of becoming aware of this determinism.
However, if there exists an alternate approach,
that allows the possibility of free will, then
even if it challenges the intuition is some other way,
it should be considered, for reasons of practicality alone, a preferable alternative.
%(We have to believe in free will; we have no choice---Isaac Bashevis Singer)
%kill the4 rest of the stuff in this section.
%We review in this section the similarities of the interpretation
%we have discussed
%with those already established which are difficult if not
%impossible to reconcile with our practical requirements of locality and
%three-di\-men\-sion\-ality.
%There is a long history of making the quantum world more sensible.
%Some have proposed amending to the Copenhagen a physical mechanism for the ``reduction of the
%wave packet''; no such mechanism has up to now gained wide
%acceptance. \footnote{D. Bohm, {\it Quantum Theory\/}
%(Prentice-Hall, Englewood Cliffs, 1951), pp. 611-23. Some of the work cited here, as well
%as an elaboration of the terminology and concepts within this section are contained
%in {\it Quantum Theory and Measurement\/}, J.~A.~Wheeler and W.~H.~Zurek (eds.),
%Princeton University Press, Princeton, 1983.} others have appealed to
%The well-known quantum mechanical
%interpretations are
%for one reason or another unimplementable
%by a cellular automaton approach
%although we shall
%see in retrospect that the interpretation we propose
%has similarities to each of them.
%%put this later on?
%The Copenhagen Interpretation
%avoids specifying a physical mechanism
%of how the measurement process. Although
%there has been a good deal of speculation
%about, for example, physical mechanisms that may explain
%the ``collapse of the wave function,''
%such mechanisms lie outside the Copenhagen
%interpretation itself.
%
%The Bohm interpretation\footnote{D. Bohm, {\it Phys. Rev.\/}, {\bf 85},
%166 (1952), and {\bf 85}, 180 (1952).}, although it evokes
%hydrodynamical picture so that at first glance, it
%might seem to be suitable for our purposes, is also beyond
%the pale of cellular automaton methods,
%in that the explanation of such nonlocal
%effects highlighted by
%the EPR paradox
%are given as the outcome of a local dynamics
%in a configuration space whose dimension is in
%general much greater than three.
%
%The ``many-worlds'' interpretation\footnote{H. Everett III,
%`\,``Relative State'' Formulation of Quantum Mechanics',
%{\it Reviews of Modern Physics\/}, {\bf 29}, 454-62 (1957).}
% is likewise
%too dificult to implement on a three-di\-men\-sion\-al
%lattice, unless one can find some method of
%simulating all the ``branching'' states simultaneously
%on such a state. We shall see that the interpretation we propose here
%does have some similarities to the many worlds
%interpretation in that it, too, allows for many
%states to be associated with a single wavefunction.
%%although as we shall
%%see, the present interpretation does have some
%%notable similarities, in that many different
%%pictures of the past and present, (i.e., many
%%different ``worlds'',) will be present in one system,
%%all of which probabilistically satisfy our
%%conceptions of quantum physics.
%%
%Other hydrodynamical models, akin to
%the ones propsed by Madelung and
%DeBroglie, seem to be incompatible
%with the experimental results
%that show quantum mechanical violations
%of Bell's inequality.
%Likewise, the field of stochastic electrodynamics,
%which can be used to obtain quantum mechanical
%results in a purely classi\-cal framework,
%has not yet been shown to contain an explicit mechanism to account for
%the long-range correlations such as those of the abovementioned EPR expermint.
%Although one could in principle
%use the leeway granted by present
%experimental uncertainties and
%by so called ``loopholes''\footnote{D.~M.~Greenberger (ed.),
%{\it New Techniques and Ideas in Quantum Measurement Theory\/},
%Vol. 480, Annals of the New York Academy of Sciences
%New York, 1986; and references within.} in order
%to make some model that
%has at least not yet been disproven,
%the models proposed thus far
%are at best {\it ad hoc\/}, and at worst,
%spookily conspiratorial.
%%the DeBroglie
%%and Madelung hydrodynamical
%%interpretation will also not do,
%%for reasons to be explained below.
%%(Wigner-"bilocution"--mention
%%that he doesn't phrase it in this way.
%
%It is of course possible to appeal to
%an interpretation that involves absolute
%determinism. Correlations between superluminally
%distant events such as are implied by violations of
%Bell's inequalities, could be explained
%by claiming that {\it all\/} events, including
%their observation by a measuring device, are
%predetermined by the boundary conditions of
%the universe to occur in the way they do. However,
%an explicit mechanism explaining how this happens, in
%a way that might aid us in our implementation, has
%yet to be proposed.
%
%Another problem with models that
%make the observed order contingent on
%the predetermined behavior of the observer is
%the loss of free will. Again this is more
%than just a philosphical conceit; it is a
%property that is enormously useful, even necessary.
%We cannot hope to simulate systems interacting
%with arbitrarily varying external fields and forces, perhaps subject
%to the outcome of events which
%they produce and which are thus initially unknown,
%if everything in the future and past must already contained in
%the boundary conditions.
%To repeat what we have said before in another context, we cannot expect that the nature of things allows us
%to simulate everything just as we want; however, if there exists an alternate approach,
%that allows the possibility of such free will, then
%even if it challenges our intuition is some other way,
%we should consider it a preferable alternative.
%%(We have to believe in free will; we have no choice---Isaac Bashevis Singer)
%
%\section{A closer Comparison With Other Interpretations}
%
%We again note in retrospect the similarities and differences
%between the local interpretation and the other conventional ones.
%Unlike the Copenhagen interpretation, there is no dividing line
%between the microscopic and macroscopic realms; the laws of
%one are the laws of the other. Even so, our modes of perception
%are such that they deal with phenomena that are averages over enormous
%numbers of microscopic phenomena. It is in contrast to such
%events that the microscopic world seems so unintuitive.
%
%As in the Bohm interpretation, there is a tangible
%space in which particles evolve, but there, one
%explains the paradoxical outcomes of measurement theory
%only by assuming that even macroscopically separated
%events are instantaneously linked together, in
%a configuration space of large, even infinite dimension. In our interpretation,
%it is the freedom made possible by
%the introduction of negative probabilities, that is used to account
%for the unusual features of quantum mechanics.
%%There is no pilot wave or guiding field either; the particle
%%dynamics, along with the cancellation of events, suffice to
%%explain the ensemble behavior that seems to indicate such
%%underlying phenomena.
%
%As in the many-worlds interpretation, each simulation
%of a universe provides a large number of
%``worlds'' (or pictures, or histories). Note
%however, that not all possible outcomes are included
%in any one simulation, and that the dynamical behavior is
%independent of the history it produces, unless we
%choose to make it otherwise.
%
%Fineally, we note that we can even make our
%theory absolutely deterministic by suitably
%defining our random number generator, although
%in this interpretation, external observers
%retain the option of manipulating
%phenomena even as the phenomena are
%unfolding.
%%*---we yield many (but not all) worlds at once.
%%*---We can recover the absolute det. prospect if we wish.
%%*A given model state may yield radically
%%*different past histories depending on
%%*how the history is started
%%*(IE, A MANY WORLDS-INTERPRETATION IN ONE)----a situation
%%*familiar to musicologists and
%%*phonologists---a sound of
%%*a given musical timbre
%%*may be manipulated to
%%*sound like one of a variety of
%%*strikingly different instruments
%%*(reeds, horns, strings, etc.), depending
%%*on the spectal characterisitics of
%%*the initial part of the waveform,
%%*i.e., the ``attack'' of the note.
%%*{\tt Mention that all sampled \break
%%*voices sound like ``ahhs''\break
%%*when the attack portion is deleted\break
%%*regardless of the vowel the \breaksinger intended.}
%*****************************************************
\section{The need for a localist interpretation }
As mentioned before, when it comes to actually simulating
physical phenomena in detail, locality becomes more
than a philosophical or aesthetic preference; it becomes an abject necessity.
In considering what has been presented here, it is
important to keep in mind that
the results, however novel they seem at first sight,
are straightforward extensions of this quite mundane
and practical desire to retain a workable concept of
locality in a three-di\-men\-sion\-al space, as well as
some freedom of will on the part of the experimenter.
In retrospect, the interpretation
%will
can then be
%then become far less radical, and can
seen as simply a natural
extension of the generalized probabilities that have
so long been in use.
The Ptolemaic model of the universe
was discarded partly because in order
to sustain it, one was obligated to
posit epicycles and highly convoluted
motions of the planets, all of which
could be greatly simplified by adopting
the Copernican model. However,
there was a trade involved, requiring
what at that time seemed a heavy price:
for this added simplicity in the
physical interpretation, we had to
give up the cherished notion
that the earth was the absolute
center of the universe.
In quantum mechanics, there is not, at the
present, any method for explaining away
all of its nonintuitive features. However,
there remains the choice of how those nonintuitive
features are manifested. Again, the choice depends on what one
is willing to trade away. It is argued here that locality, being
so practical and useful,
is a feature that should not be forsaken if
another alternative exists, even if that other alternative
contains nonintuitive features of its own.
% (such as negative events, and the relativity of histories).
It is argued here that the benefits of this model are such as to make it
%Because the notion of locality is retained, the present model is
the simplest choice overall.
If, by way of negative events and the nonabsoluteness
of histories, we choose to give up our cherished
notions concerning the past, we can retain a much more workable
way of dealing with the present.
I wish to thank
Tommaso Toffoli
and all the members
of the Information Mechanics
Group at MIT, whose help
and guidance made this
work possible. I am grateful to Charles H. Bennet for his
helpful suggestions.
Support was provided in part by the National
Science Foundation, grant no. 8618002-IRI,
and in part by the Defense Advanced Research
Projects Agency, grant no. N00014-89-J-1988.
\vspace{.6\baselineskip}{\obeylines
\noindent {\large \bf References}\nopagebreak
\vspace{.3\baselineskip}\nopagebreak
}
%
\noindent$^1$H.~Hrgov\v ci\'c, ``Discrete representations of
the $n$-dimension\-al wave and Klein-Gordon equations'', preceding article.
\medskip
\noindent$^2$A paper by J.~S.~Bell,
``Beables in Quantum Theory'', in B.~J.~Hiley and F.~David Peat (eds.),
{\it Quantum Implications\/} (Routledge \& Kegan Paul, London, 1987), presents related considerations.
\medskip
\noindent$^3$H. Poincar\'e, {\it Science et M\'ethode\/}
(Ernest Flammarion, Paris, 1912),
Livre II, Chapitre I, p.~94.
\medskip
\noindent$^4$M.~Hillery,
R.~F.~O'Connell, M.~O.~Scully, and E.~P.~Wigner,
{\it Physics Reports\/} (Review Section of Physics Letters) {\bf 106}, 121 (1984);
P.~A.~M.~Dirac, {\it Reviews of Modern Physics}, {\bf 17}, 195 (1945);
R.~P.~Feynman, `Negative Probability' in B.~J.~Hiley and F.~David Peat (eds.),
{\it Quantum Implications} (Routledge \& Kegan Paul, London, 1987).
\medskip
\noindent$^5$The ``Negative Probability'' paper of Feynaman
cited above, contains several elementary examples illustrating this point
with respect to the Wigner function, for which the same can be said, and shows
how negative probabilities can be used to generate
the distributions of particles arising in such seemingly
nonlocal phenomena as the EPRB experiments.
\medskip
\noindent$^6$See for instance, N. F. Mott's explanation of
$\alpha$-ray tracks from purely a wavelike perspective, in
J.~A.~Wheeler and W.~H.~Zurek (eds.), {\it Quantum Theory and Measurement\/}
(Princeton University Press, Princeton, 1983).
\medskip
\noindent$^7$M.~Born, in
{\it Nobel Lectures in Physics\/} (Elsevier Publishing Company,
Amsterdam, 1936), p.~256.
\medskip
\noindent$^8$For example, Chapter III, Wheeler and Zurek, cited above.
\medskip
\noindent$^9$D. Bohm, {\it Quantum Theory\/} (Prentice-Hall, Englewood Cliffs,
1951),
pp.~611-23. Some of the work cited here, as well
as an elaboration of the terminology and concepts
within this section can also be found
in Wheeler and Zurek, cited above.
\medskip
\noindent$^{10}$H. Everett III,
``\,``Relative State'' Formulation of Quantum Mechanics'',
{\it Reviews of Modern Physics\/} {\bf29}, 454-62 (1957).
\medskip
\noindent$^{11}$See, for example, M.~Jammer, {\it The Philosophy
of Quantum Mechanics\/} (John Wiley \& Sons, New York, 1974), Chap.~2.
\medskip
\noindent$^{12}$D. Bohm, {\it Phys. Rev.\/}, {\bf 85},
166 (1952), and {\bf 85}, 180 (1952).
\medskip
\noindent$^{13}$D.~M.~Greenberger (ed.),
{\it New Techniques and Ideas in Quantum Measurement Theory\/}
Vol. 480, Annals of the New York Academy of Sciences
New York, (1986), and references within.
\end{document}
\section{Appendix}
IN THIS SECTION WE GO REVIEW SOME OF THE TERMS FOUND IN THIS PAPER.
EXTERNAL VS. INTERNAL
INTERPRETATION
MODEL
MEAURING DEVICE (SINGLE AND MULTI) AND MEASUREMENT.
INCLUDE IN THIS PROBE, INTEGRATION PATH, COUNTER, OR COUNTER READING,
RUN. IN SINGLEPARTICLE CASE, COUNTER INCREMENTS WHEN
IT SEES A BRA-KET PAIR OF THE FORM ++ OR -- AND DECREMENTS
FOR THOSE OF THE FORM +- -+. (WE CAN AFFIX A NATURALIZER
FUNCTION ONTO THE COUNTER WHEREBY THE COUNTER READING IS
RECTIFIED SO AS TO BE NONNEGATIVE, BUT FOR MACROSCOPIC
MEASUREMNTS, THIS IS UNNOTICEABLE. ALSO NOTE ITS
LINEAR, SO THAT TWO + BRAS AND THREE KET PARTICLES IN ONE
PLACE COUNT FOR SIX. FOR A MULTI-EVENT,
MAKE THE COUNT POSITIVE IF THE KET TAGS ARE A POS PERMUTATION
OF THE BRA TAGS, NEGATIVE IF THEY ARE A NEG PERMUTATION AND
ZERO OTHERWISE. NOTE THAT AT ANY GIVEN POINT IN SPACE,
ALL KET PARTICLES HAVE SAME TAG, AND ALL BRA PARTICLES
HAVE THE SAME TAG.
(NOTE THAT IN ORDER FOR A MEASUREMNET TO BE PERCEPTIBLE TO HUMANS
IT NEEDS TO CORRESPOND TO A LARGE COUNTER READING, WHOSE
XACT VALUE (IE DOWN TO THE LAST SIGNIFICANT DIGIT) IS UNIMPORTNT.)
PICTURE-HISTORY
\section{Appendix}
In order to adapt the work of our previous paper to quantum mechanics
we need to elaborate the concepts within. In particular, we must
show how we can adapt our work to yield particle behavior that is truly
localized (for given the fact that even a single particle wavefunction is simulated
through branching process involving large numbers of particles, it remains
to be seen why multi-particle correlations, such as simultaneously
capturing a particle in both of the transmission regions
of a half-silvered mirror, are not observed for one-par\-ti\-cle states). Also, we will
show how multi-particle events, including their associated statistics,
can be trivially added by way of this model.
As given in the previous paper, the wave function of a particle
that obeys, say, the three-di\-men\-sion\-al Klein-Gordon equation (on an
orthogonal lattice of unit spacing) is at any point an
algebraic sum of a number of component ``traveling wave'' functions that are first order in
time, and that are defined on the links between neighboring points on
the lattice. The points of the lattice then serve as a kind of
black box---a particle traveling along a link leading into
a given lattice point, may result in several particles leaving,
with the phases of the particles being such that the algebraic
mean of the flow entering or exiting a lattice point at a certain
time is given by the Klein-Gordon equation. By a flow in
a given link, then,
we mean the number
$$f_{z+}^{\rm in}({\bf x};t) = {1\over{\cal N}}\bigl(E(N_+) - E(N_-) + $$ %\right.$$
%$$\left.
$$ i E(N_i) - i E(N_{-i})\bigr) $$
where $E(N_{\sigma})$ is the expected number of particles
of the respective type that would be found in the corresponding link
(here we have assumed it is the link leading into the
point at ${\bf x}$ along the positive $z$ axis), if the
simulation were to be repeated a large number of times ${\cal N}$, or alternatively,
if the boundary conditions corresponding to the phenomena in question were
to be repeated in different regions of one large simulation.
(Note that we have ommitted the dependence of the
terms on the right hand side, save for ${\cal N}$, on
${\bf x}$ and $t$.)
We also showed that in order to obtain the Klein-Gordon
equation, the incoming flows are modulated
by a factor of $e^{\pm i\alpha}$ at every lattice. That is,
given a net unit flow into any link the outgoing links
are multiplied by $e^{\pm i\alpha}$, where $\alpha$ is
a term proportional to the mass of the simulated particle,
in comparison to the situation when the mass is zero.
This means that we could implement the interaction by
adjusting the branching ratios so that on the
average there are $\cos\alpha$ particles in the positive state
and $i\sin\alpha$ particles in the positive (or negative) imaginary
state that leave a lattice point, for every positive particle
that enters there, and so on. (We could of course, have
altered our model to allow particle to have an
arbitrary complex argument, that is,
possessing a continuous magnitude and
phase, and
then recover integrality by fiat
in the counting protocol,
but for present purposes,
the above approach is satisfactory.)
Note that in the formalism we have given, the measuring process
yields inherently integral results. As mentioned above, this
does not yet insure, given the branching dynamics we have proposed,
that a one-particle state does not imply
more than one particle present at a given time. One removes
the notion of bilocution
by modifying the dynamics so that particles that
are ``siblings'' of some branching phenomena are correlated in
such a way as to make any self-correlation seems to be zero (i.e.,
make it seem as if there is only one particle at a time associated with
any single particle state).
%\footnote{As Wigner has shown (reprinted
%in {\it Wheeler and Zurek}, above, pp. 310-312), a single particle found at one
%point at some given time, has a nonzero probability of being
%found even at spacetime points that lie outside the light cone focused
%at the first spacetime point, so that
%in some situations, a particle can (in a suitable reference) frame
%appear to be in two places at once. The formalism we will give
%can be easily modified to account for this (something which cannot
%be said about some other interpretations, but for now we will ignore such
%complications.}
A few preliminary definitions will make these concepts easier to explain.
Consider defining a weight function, $w\{{\bf x}_i\}$
on any particular assignment of particles on a lattice,
which is 1, 0, or $-1$, according to whether
there is, respectively, a positive particle, no particle, or a negative particle
at the point ${\bf x}_i$. Consider next the
hodotic solution of the discrete two-di\-men\-sion\-al hodotic solution
described in the previous paper, corresponding to a nonzero
flow emanating from the origin along the positive $x$ axis
(cf. (Fig.~\ref{fi:x+-rep}), reproduced from the previous paper.)
\begin{figure}
\rule{.2pt}{13em}
%Have a diamond, with the coefficients ${1\over2}$ on
%home plate, first and second base, with $-{1\over2}$
%on third base. If clearer, make the drawing three-di\-men\-sion\-al
%with time along the vertical axis.
\caption{The $x+$ hodotic solution, at $t=2$.}
\label{fi:x+-rep}
\end{figure}
Obviously, the assignments of particles on a lattice
that we use to simulate this initial configuration should
be such that the mean over all such assignments (at time $t=1$)
of the weight function $w\{(0,0)\}$ is 1.
Furthermore,
the mean over all the assignments in the subsequent time step will be
$1\over2$ for the weight function $w\{(0,1)\}$, $-{1\over2}$
for the weight function $w\{(0,-1)\}$, and so on. Likewise,
in any run of simulations corresponding to the wavefunction
$\psi({\bf x},t)$, the mean at the respective time $t_p$ of the weight function
$w\{{\bf x}_p\}$ will simply be $\psi({\bf x},t)$.
One can also define weight functions for two or more points.
For example, $w\{{\bf x}_i;{\bf x}_j\}$
is simply equal to
$w\{{\bf x}_i\} w\{{\bf x}_j\}$, if the state of
a particle exiting a given lattice point at the given time depends solely on the
particles that entered there in the previous time step, but it may be different,
if the emission of particles emmited from a point in one direction is correlated
with the emission of particles emmitted in another.
For instance, let us suppose that
for every time a particle enters a node along one positive axis in such a
way that a particle is emitted along the positive direction along an orthogonal axis, we prescribe the
dynamics so that a particle of the same sign is also emitted only along the
negative direction along that orthogonal axis axis. To give an example, whenever
a (positive) particle flowing into a lattice point along the $x+$ direction results
in a particle exiting (in the positive state) along the $y+$ direction, we arrange
that there is always also a particle exiting (in the positive state)
along the $y-$ direction.
Then, given again that the wavefunction
under consideration is the above two-di\-men\-sion\-al hodotic solution,
we can see see at time $t=2$, the mean value of
the weight function $w\{(1,0);(0,1)\}$ is zero, while that of the weight function
$w\{(0,1);(0,-1)\}$ is one, even though the products of the means for corresponding
single point weight functions is of course equal to the product of the amplitudes at those
points, which is $1/4$.
%Note that we have up to now assumed that the
%amplitude of the outgoing traveling waves was given by particles all of the same sign.
%For example, the amplitude of $1\over2$ at the point $(1,0)$ corresponding to
%the above hodotic solution came about
Even though the dynamics given for the wave functions
is that of a branching process, whereby
one particle at a given time can mean several
particles later on, one would still like
the {\it net\/} value of the quantity $w({\bf x}_i,{\bf x}_j)$
to be 0, just as it trivially would be if there were only
one particle on the lattice (per one-par\-ti\-cle state) at any given time. Then along with
our protocol, we would see that macroscopically speaking, the phenomenological outcome of such a dynamics is
the assumption that quantum phenomena are inherently discrete,
i.e., the ``branching'' aspect of the phenomena cancels away so as to become unnoticable.
This does not mean that when we simulate multi-particle states
that all interparticle measurements will vanish, for
in general, the wavefunctions for, say any two independent particles
will be totally uncorrelated, and therefore their two-point weight functions $w\{{\bf x}_a;{\bf x}_b\}$
will (assuming that the wave functions of the two states involved are not both nonzero at either
${\bf x}_a$ or ${\bf x}_b$,)
simply be equal to
the product of the corresponding single particle weight functions. We defer more complicated choices until
we discuss multi-particle states in greater detail.
Consider again, our two-di\-men\-sion\-al hodotic solution of
the discrete wave equation (Fig.~\ref{fi:x+-rep}).
As mentioned above, a particle entering the origin, say, in the positive state and along the positive $x$ axis
must in the subsequent time step lead to a mean value (when averaged
over all possible subsequent configurations of particles on the lattice)
of ${1\over2}$, ${1\over2}$, $-{1\over2}$
and ${1\over2}$ for the respective weight functions $w\{(1,0)\}$, $w\{(0,1)\}$, $w\{(-1,0)\}$, and
$w\{(0,-1)\}$. Let us for convenince restrict ourselves to all assignments
that have nonzero values for the weight function $w\{(1,0);(0,1)\}$, i.e.,
all configurations which at time $t=2$ result in a particle at
both of the points $(1,0)$ and $(0,1)$; this will make the mathematics
simpler.
Let us further suppose, for the purposes of
this example, that
in addition to the four ``algebraic'' states (positive, negative, etc.)
mentioned above, a particle has an extra
degree of freedom via four substates labelled by the indices
$\{ x+, y+, x-, y-\}$. Let us suppose that
when counted by a measuring device, the sign of the contribution
of that particle is indicated by its algebraic state, unless
the particle has a substate whose index indicates the direction
opposite to the one in which the particle is traveling at the time in question, in
which case the contribution to the counter is the negative of what is
indicated by the algebraic state. Therefore, a po\-si\-tive-ima\-gi\-na\-ry
traveling along the $y-$ direction
participates in the counting protocol as if it were a particle having a sign of $+i$, unless
its substate is of the type $y+$, whereupon it participates as if
it were a particle of sign $-i$.
Therefore, where in previously considering the above situation an amplitude of
$+{1\over2}$ meant that in an ensemble of such situations, there would be
a positive particle at the corresponding point in one-half of the ensemble
members, we now have the case that there may be particles of different
signs, and that only the net contribution of all the situations is $+{1\over2}$.
This of course means that if the sum over the four substates
representing at $t=1$ the flow to the point $(1,0)$ is
to be $1\over2$, etc., there must with probability 1 be a particle of
some kind at any of the four points surrounding the origin, given that
there was a particle at the origin at $t=0$; only then
can the net amplitude be of the appropriate magnitude. Considering the situation
at just the points $(1,0)$ and $(0,1)$, if we specify that
that the particles resulting at these two points from a
previous particle arriving at the origin along the $x+$ direction, must never be in the same
substate.
All other choices for the pairs are equally likely.
If we sum over the amplitude contributions for all the possible choices,
we will see that the mean value of the weight function $w\{(1,0);(0,1)\}$
is indeed zero. It is important, when simulating wave functions
whose inital values are spread over a large number of lattice points, that
all the cross-correlation of the initial values are also zero; such will ofcourse, trivially
be the case if we initially start all simulations with just one particle per state.
%Note that we assumed that the proper way to simulate the initial event
%(i.e., the amplitide of 1 at the origin) was by lattice configurations for
%which there was always one positive particle initially at the origin. We might have
%also used a number of positive and negative particles whose mean value was one.
%Such a complication becomes necessary if we otherwise find that
%the desired probabilites of all events resulting from a unit flow at the origin, sum up
%to a number greater than 1.
THE STUFF BELOW IS STUPIDLY INCORRECT. IGNORE IT
Let us next consider the simulation of multi-particle phenomena, in particular
the notion of exchange symmetry, fermions are characterized by the
Pauli Exclusion Principle, i.e., the
property that no two identical particles may be in the same state at any given
time.
In the previous paper, we have shown that wave solutions may be written as
a sum of terms, each of which represents a random walk on a lattice.
Multi-particle fermion events, say, those corresponding to a particle that
was initially at point ${\bf x}_a$ and one that started at the
point ${\bf x}_b$, with the corresponding states being
labeled as $\psi_a({\bf x})$ and $\psi_b({\bf x})$, can then be written as a sum over all
possible pairs of such paths, except those that imply the
two particles both travelled to the same spacetime point.
(Note this refers only to particles that simultaneously
{\it enter\/} a given point. This does not preclude a
branching process, which implies that more than one
particle may exit a given point.)
Such an exception is trivial to introduce into our dynamics:
wherever more than one particles meander into the same state
and same link, we annihilate all but one of them; the branching will ensure that on the
average, the number of particles does not decrease.
It is much more difficult to
keep track of this constraint {\it mathematically}; however, it
turns out that this is precisely what exchange
symmetry does for us.
To see this, let us rather inconsistently
{\it pretend\/} that the two indistinguishable
particles could be distinguished enough to label one
as particle 1 and the other as particle 2.
Next consider the expression
$$\psi({\bf x}_1,{\bf x}_2,t) = {1\over\sqrt2}\bigl(\psi_a({\bf x}_1)\psi_b({\bf x}_2)
- \psi_b({\bf x}_1)\psi_a({\bf x}_2)\bigr)$$
vis-\`a-vis a summation of terms, each consisting of a product of pairs of paths.
By a collision, we mean the simultaneous presence of
two or more particles at a {\it link}, rather than merely two or more
particles at a point.
Paths implying a collision between the particles appear in both
of the products of the left hand side of the above equation; if the
situation were two-di\-men\-sion\-al, the contribution
of the product in one of the terms lends itself to seeing
the two particles as ``bouncing'' off one another, while the other
suggests the view of the two particles somehow going through one another (Fig.~\ref{fi:mul-par}).
\begin{figure}
%Show two sets of two pictures in the first, show schematically the contribution
%of \psi_a \psi_b and \psi_b \psi_a for nonintersecting paths, and in the next two sets,
%the contribution from intersecting paths.
\rule{.2pt}{13em}
\break
\rule{.2pt}{13em}
\caption{Amplitude contributions from noncolliding paths (top) are
disinct, while those of colliding paths (bottom) are not, so
that the latter are cancelled in the antisymmetrization process.
%(Contrary to appearances, the top left picture
%represents a pair of paths for which the particles do {\it not\/} collide.)
}
\label{fi:mul-par}
\end{figure}
In either case, however, the sign of the contribution is the same,
so that it vanishes in the antisymmetrization process. (The
factor of $1/\sqrt2$ is likewise a reflection of the supposition
that particle exchange leads to a distinct rigion of phase space.)
Therefore, we see that according to our formalism,
the concept of particle exchange is merely a mathematical
artifice that allows us to correctly account for the %
Pauli Exclusion principle.
%Let us suppose that the universe we wish to simulate consists
%of a large number of boxes, with each box equipped with a
%clock and coordinate system. Using the terminology of
%the previous paper, let us suppose that every box contains
%at its respective initial time a single particle whose
%wave function is that of a hodotic solution. It is clear that,
%given the fact that the amplitude and amplitude squared
%are one along a single link within each box, each box
%then initially contains a single bra particle and ket particle
%travelling along the corresponding link. The state of the
%bra and ket particles in a given box can be positive, negative, etc., but
%it must be the same for the bra and ket particles of a given
%box. At any subsequent time, by placing our counter at the
%point that in each box is labeled by ${\bf x}$ and $t$, we will
%obtain for a large number of boxes, a result consistent with
%quantum mechanics, provided we follow the given protocol.
%Suppose that the wave function in each of the boxes was
%no longer merely one hodotic solution, but rather, a mixture of
%two, say, $$\psi({\bf x}, t) = {1\over\sqrt2}(h_{x+}({\bf x-x_a}) - h_{z-}({\bf x-x_b})$$
%where the dependence of the $h_{\sigma}$ on time has been suppressed for
%convenience. In each box, at each of the points ${\bf x_a}$ and
%${\bf x_b$ we generate a random number from a uniform distribution
%ranging from 0 to 1. If the number is less than or equal to
%$1\over\sqrt2$ we place a ket particle at that point. We then
%repeat the entire process for bra particles; all of the assignments
%are made independently of one another---we shall discuss how to
%resolve the difficulties this introduces below. For now, we
%see that in any of the boxes the point ${\bf x_a}$ may contain
%a bra particle or may not, likewise for a ket particle, and
%likewise for the point ${\bf x_b}$.
%Although the choice of phase for the particles
%is again arbitrary so long as the bra and ket particles in any box
%are the same sign, the choice of signs must be consistent throughout the
%box. In accordance with the above boundary conditions, if we agree
%that the particles of some box at the point corresponding to
%${\bf x_a}$ are to be
%in the positive state, then the particles at ${\bf x_b}$
%must be in the negative state. The generalizations to
%other combinations of hodotic solutions, and therefore
%arbitrary wave functions, should now be apparent, although
%in many cases, initial configurations in which all
%the particles are in wavefunctions corresponding to hodotic solutions
%will suffice.
%INCLUDE AN APPENDIX: DESCRIBE IN THIS MODEL, HOW SEVERAL TRAVELING WAVES ARE NEEDED TO
%GET ONE SCALAR WAVE; DESCRIBE IN MORE DETAIL HOW THE MODULATION FACTOR COMES ABOUT
%(WE USE ALPHA SUCH THAT COS ALPHA AND SIN ALPHA ARE RATIONAL AND HAVE TO MAKE THE
%N-PLICATION FACTOR CORRESPONDINGLY LARGER. DISCUSS HOW TO SIMULATE
%1) BASIC HODOTIC SOLUTION (FOR 1 PARTICLE)
%2) BASIC SUM OF TWO OR MORE HEODOTIC SOLUTION (1 PARTICLE)---SHOULD IMMEDIATELY
%GENERALIZE TO AN ARB. LIN. COMBINATION OF HODOTIC SOLUTIONS
%3) NOTE THAT EVERYTHING IS ASSIGNED INDEPENDENTLY OF EVERYTHING ELSE.
%NOTE THAT THIS CREATES PROBLEMS---EG, GETTING TWO PARTICLES PER BOX IN
%A ONE PARTICLE SIMULATION----BUT NOTE THAT IN AN N-PARTICLE SIMULATION
%UNIVERSE (N IS HUGE) THE SMALL-N CORRELATIONS ARE OK. THIS IS AS IT SHOULD
%BE; DON'T EXPECT THAT THE BIG PICTURE IS `PROBABILISTICALLY DETERMINED''
%FOR WE HAVE ONLY ONE OF THEM.
Imagine a three-di\-men\-sion\-al
orthonormal lattice whose
whose evolution is given at
discrete and equal intervals of
time, but whose lattice (and time)
spacing is far too small to be
measured with present-day methods.
On this lattice are certain number
of particles performing a type of
random walk to be discussed below.
Each particle is one of two kinds,
which we shall distinguish as the
``bra'' and ``ket''. These qualities
are unchanging features, so that
bra particles never turns into ket particles
and vice versa. A particle can, however,
be in one of four states, that can
be likened to algebraic signs in a
complex space, which we shall therefore
label as positive ($+$), negative ($-$),
positive imaginary ($+i$) and
negative imaginary ($-i$).
At any moment, a particle has a
probability of branching into
several independent copies of itself.
Also, if two particles of
the opposite ``sign'' land on
the same lattice point, e.g.
a positive imaginary particle
meets a negative imaginary particle,
they annihilate.
Let $f_{\sigma}({\bf x},t)$, where
$\sigma \in \{ +,-,+i,-i\}$,
be the expected fraction of
the ensemble giving the number
of systems with exactly one ket particle
in the $\sigma$ state at
${\bf x}$ and $t$, plus
twice the fraction of the systems
that have exactly two such particles,
etc., and
let $f^*_\sigma({\bf x},t)$
give the corresponding fractions
for the bra particles.
Note that a term of the form
$f^*_{+}({\bf x},t)f_{-}^2({\bf x},t)$
then gives the expected fraction of
the ensemble for which there
is a collision of a positive
bra particle and a negative
ket particle at ${bf x}$ and $t$. {\tt True?}
Let us suppose that the
dynamics and initial distributions
of these particles are such that
for some large {\it ensemble\/} of
identical systems,
$$\psi({\bf x},t) = k\left( f_{+} - f_{-}\right.$$
$$\left. +if_{+i} - if_{-i}\right)$$
and
$$\psi^*({\bf x},t) = k\left( f_{+}^* - f_{-}^*\right.$$
$$\left. +if_{+i}^* - if_{-i}^*\right)$$
where the dependence of the probabilities
on space and time has been omitted for
convenience, and where $\psi^*({\bf x},t)$
is the quantum mechanical
wave function for a electron, whose
initial values are such that the above
equations are true at time $t = 0$.
Let $k$ be such that
$$\left|\psi^2({\bf x},t)\right|{a^3 \over k} =
\sum{\sigma, \sigma'} \hat f^*_{\sigma}\hat f_{\sigma'}$$
where $\hat f_\sigma$ and $\hat f^*_\sigma$ are
the appropriate fractions multipled by 1, $-1$, $+i$,
or $-i$, according to the sign associated with each
fraction. (We assume that the left hand side of
the above equation, when summed over the
three-di\-men\-sion\-al region we are considering, is 1.)
By a {\it measurement}, we shall mean the following:
take a measuring device from
In the model we are proposing, we shall assume
as given a three-di\-men\-sion\-al space, say a
box, which is closed to the outside world,
and in which are a certain number, $N$, of particles.
For the moment, we shall assume these are
all mutually interacting electrons, whose energy
is much lower than that required for
electron-positron pair production,
so that $N$ may be taken as fixed.
The evolution of this system may be
given completely by the
A specification of all
or part of our space, i. e. a {\it measurement\/},
will be given by
specifying at some time a closed Gaussian surface
as well as the number of {\it electron events\/}
within that surface at the given time.
{\tt DETERMINING NUMBER OF EVENTS IS DONE BY INTEGRATING (SWEEPING)
OVER ALL SPACE AND COUNTING THE
NUMBER OF "EVENTS" AT EACH LATTICE SITE, SO THAT ACTUALLY
WE ARE CONSIDERING A REGION IN SPACETIME}
By manipulating the surface, we may ultimately
describe anything we want about the space.
For instance a measurement might consist
in asking at some time, how many
events are there in the right side of the box,
how many events are there in the region immediately
behind the calcite crystal corresponding to
a ``spin up'' polarization
in which all information is ultimately
to be dervied fromTo begin the rules of the game,
state that all information is to
be reconstructed from density of
fermion events, this being more than
enough to classically specify our system.
??We may even go so far as to only have electrons.
By fermion events we mean the following---blah?
--- any sweep by a Gaussian surface will
encounter a certain number of events--
think of a beam of particles, in the
reast frame of the beam, so that the detector
sweeps past them. Up to now, everything
has been positive--We shall allow for
events to negative as well--