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<h2>Section Four<br>
<span class="section-subhead">Back Through The Looking Glass</span></h2>
<p>This final section of the book contains the more speculative writing I
have been alluding to in previous sections. If I can be confident of
anything regarding what follows, it is that I will present ideas that are
just plain wrong. But history shows that even wrong ideas can be
interesting and thought-provoking; sometimes they lead to some kind of
progress. If there is a central theme to this section, it is that neither
relativity nor quantum mechanics is complete without the other, and that
their connection is much deeper than most people suspect: one seems to be
the reflection of the other. In my view, relativity gives an
almost-but-not-quite explanation of why things fall down, and the final
piece of the puzzle is to be found in quantum mechanics; conversely, the
only satisfying answer to the mystery of entanglement seems to lie in
relativity's concept of the spacetime interval. And here I examine a
possible answer to the questions regarding entanglement and locality that
continue to generate hundreds (if not thousands) of pages of books and
articles each year.</p>
<!--
**********************************************************************************************************
**********************************************************************************************************
******************************* CHAPTER 32 ******* **************************************************
**********************************************************************************************************
**********************************************************************************************************
-->
<hr>
<div id="Chapter32">
<h3 id="MappingSpacetime">Chapter Thirty-Two <br>
Mapping Spacetime</h3>
<p>At the end of Chapter Twenty-Seven, I concluded that our notions of
locality and causality needed to be re-examined in view of quantum
interference and entanglement. And I have an idea to propose in that
regard, one that almost makes sense in regard to photons and that may be
more difficult to accept with regard to electrons. It seems too simple and
obvious for everyone else to have missed it, and too interesting to have
been dismissed without comment, so in that sense it's mysterious to me.
Why have I never seen it mentioned anywhere else, if only to point out
that the idea is wrong?</p>
<p>There's something about Maxwell's equations that is easy to miss unless
you are looking for it. It seems as though they have been urging us to
consider the world in less local terms all along. None of those equations
describe conditions at a single event or <em>point</em> in space. They
are written in terms of the curl and divergence of a vector <em>field</em> (in
their differential forms), relating conditions among neighboring points,
or in terms of that field's flux through the surface of a <em>volume</em> or
intensity along a closed curve bounding some <em>area</em> (in their
integral forms).</p>
<p>On that note, do you realize that the energy associated with an
electrical circuit doesn't flow in the wires and current itself, but in
the field surrounding and permeating the circuit? <a href="https://www.youtube.com/watch?v=bHIhgxav9LY"
target="_blank">Veritasium</a> did a pair of thought-provoking videos
on this. We see this principle in action all the time, even though we
don't often think about it. Many of our mobile phones and electric
toothbrushes recharge by magnetic induction of current rather than by
wired connections. It reminds me of the way that electrons and photons
needn't be limited to one path or the other in a double-slit experiment.</p>
<p>Allan Adams began his Quantum Physics 8.04 lectures at MIT with a
discussion of electron spin measurements<!--https://www.youtube.com/watch?v=lZ3bPUKo5zc-->.
One surprising aspect of these measurements is as follows: A spin detector
of the sort used in the Stern-Gerlach experiments (Chapter Twenty-Seven)
can sort electrons into <q>left</q> and
<q>right</q> spin groups. We discard the
<q>right</q> group and
send the <q>left</q> group into a second detector which
measures them on a different axis of spin and thus separates the <q>left</q>
group into subgroups of <q>up</q>
and <q>down.</q> What we have already learned so
far is that if either of these <q>up</q>
and <q>down</q> groups are fed into a third
detector, the results of another <q>left-vs-right</q> measurement will be
completely random, because the properties of vertical and horizontal spin
cannot be meaningful for these electrons simultaneously. The surprising
thing is that if we recombine the <q>up</q> and
<q>down</q> groups in a way that
deprives us any record of their vertical spin, and then feed them into the
final horizontal detector, they will all be measured with their original
<q>left</q> spin. It is as if the recombination process
<q>erases</q> the
not-technically-a-measurement that took place at the vertical detector. So
we are left with no consistent or satisfying answer about the path taken
by any of the electrons in our setup. They couldn't have taken <em>both</em> the
<q>up</q> and
<q>down</q> paths out of the vertical detector, because they are
particles, aren't they? And they couldn't have taken one path or the
other, because that would mean that they demonstrated a perfectly
persistent horizontal spin even through the process of having been
meaningfully sorted on a vertical basis.</p>
<p>But what about <q>none of the above?</q> Is it possible that the electrons
don't really flow through the detectors <i>per se</i>, but like the
energy of an electrical circuit, they flow through a more
widely-distributed field? Are the detectors merely an influence on that
field? Do the electrons really exist at definite places between the times
we are measuring them?</p>
<p></p>
<p></p>
<p>Let's go back to the double-slit experiment (Chapter Twenty-Seven) and
consider how laser-emitted light interferes with itself to produce a
diffraction pattern on a photographic plate. <q>Well,</q> we say, <q>if light is
a particle called a photon and it falls on the plate one particle at a
time, can we tell which slit it went through?</q> And the answer is no.
Anything we do to measure which slit the light went through results in the
the light no longer interfering. It is as though the light must be allowed
to go undisturbed and undetected through both slits at once in order for
interference to take place. And one of the great questions that follows
this realization is: <q>how can the light at one slit know whether the other
slit is open?</q></p>
<p>For something traveling at high speed, space and time are measured
differently. As its speed approaches the speed of light, it measures all
distances between events along its path to be zero in both space and time.
In our laboratory frame of reference, the emission of the photon at the
laser is one event and the photon's detection on the photographic plate is
another event, separated in both space and time from the first. In the
light's frame of reference, these events are one. To the light beam, both
slits might be at the same place and time as the place the beam originates
and all of the places where it might land. The photon is present at all of
these events <q>at once.</q> I should mention that this view is not widely
held, at least not often discussed. There are obvious problems with
talking about a <q>frame of reference</q> that no conscious being can measure
from. John Gribbin writes in <cite>In Search of Schrödinger's Cat<!--p. 224--></cite>:
<q>It is not really clear what the concept of locality means for a photon.</q>
I don't recall ever having read anyone else addressing this problem at
all.</p>
<p>Consider the definition of the spacetime interval, specifically the "null
intervals" over which the distance between two events is exactly the same
as the time between them multiplied by the speed of light: <i>x = ct</i>.
The interval between any two events intersected by the same ray of light
is zero. The meaning of that zero seems to be that light can flow between
these events, and these events alone. Do these events have a strict time
order? Recall that timelike intervals do and that spacelike intervals do
not. <q>Light-like</q> null intervals sit on the very boundary between having
or not having a strict time order, between allowing or not allowing
causation.</p>
<p>Another potential problem with this idea of locality for all points on
all possible paths for a single photon is that special relativity
specifies that length contraction and time dilation occurs in the single
direction of relative motion. It doesn't obviously support the idea of a
particle going in multiple directions (e.g. a photon passing through two
slits) and each of these directions demonstrating equal amounts of length
contraction and time dilation. But to suppose otherwise seems to require
rejecting the idea of equality and one of the longest-standing postulates
of logical thought: if two things are like another thing, then they are
like each other. If A=B and A=C, then B=C. If there is zero separation
between the emission event and the passing-through-slit-one event, and
there is zero separation between the emission event and the
passing-through-slit-two event, can there be any separation between the
events passing-through-slit-one and passing-through-slit-two?</p>
<p><em>If</em> we stipulate that the equality holds and that the principles
of relativity apply in such a way, then the question is no longer, <q>how
can the state of one slit affect what happens at the other,</q> because they
are both the same event, or at least nearly so. The question becomes
rather <q>how is the end event (in our laboratory reference frame) for each
individual photon chosen, and what makes one event more probable than
another?</q> In other words, why do we get a dot at <em>this</em> point on
the plate rather than any of the others?</p>
<p>It is interesting to consider that a photon may take two paths (of
differing length) through two different slits to land on the same point on
the photographic plate at two different times, and thus at two different
events. From our frame of reference, the light wave may be in different
phases at those times and may thus destructively interfere. But from the
photon's frame of reference, both of those two events are simultaneous.
Perhaps there is something about the differing states of the plate itself
at those events (as measured from our standpoint) that makes it more or
less likely to interact with the photon.</p>
<p>This thought process led me to question everything I think I know about
electromagnetism. Light seems to have wave characteristics only in the
spaces <em>between</em> where it is emitted and absorbed (or reflected)
and then to take effect in particle-like ways. But from one frame of
reference (its own), the light does not travel at all: the emitter and
absorber are (very nearly?) in physical contact. There is no <em>between
</em> and therefore no need for an intervening <em>field</em>.</p>
<p>What I am proposing here is a broader definition of locality. "Here and
now" is also <q>there and then</q> for a limited set of
values of <q>there and then,</q> and with only a partial symmetry in how such events may overlap and
interact, as I will clarify shortly.</p>
<p>I ruminated on this possibility for many years before finding any hint
from a better-educated source that there might be something to it. Some of
you may be familiar with the name Richard Feynman and/or Feynman diagrams.
Feynman was a close associate of John Wheeler and a physics lecturer of
great renown. A <q>Feynman diagram</q> shows possible interactions of particles
over time and in one dimension of space.</p>
<div class="figdiv nofloat_600 float_300_66 float_1050_50 float_1200_33" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/Feynman.png" style="max-height: 300px;">
<figcaption>
<b class="figlabel">Figure 32-1</b>.
<span class="ficaption_description">
A Feynman diagram in which two electrons
repel one another by exchanging a photon. Adapted from <a href="https://commons.wikimedia.org/wiki/File:Equivalent_of_one_of_first_published_Feynman_diagrams.svg">Wikimedia
Commons</a>.
</span>
</figcaption>
</figure>
</div>
<p>In Figure 32-1, two electrons are shown moving toward one another at the
bottom of the diagram, which represents its beginning. Since both
particles carry the same charge, they must repel one another. Rather than
supposing that this repulsion occurs due to the electric field created by
both particles, we can imagine that it is accomplished by the exchange of
one or more <q>force-carrying</q> particles. In the case of electromagnetic
force, the responsible particle is the photon. In <cite>How to Teach
Relativity To Your Dog</cite>, Chad Orzel writes:</p>
<blockquote>
While it may seem strange to describe forces in terms of particles, the
notion of exchange particle mediating the interactions between physical
particles is one of the central ideas of modern theoretical physics,
particularly the branch known as quantum electrodynamics (QED). This is
most clearly described in terms of Feynman diagrams … which are nearly
ubiquitous in theoretical physics.
<!--p. 186-->
</blockquote>
<p>Midway up the diagram in Figure 32-1, the electron on the right emits a
photon which is absorbed by the electron on the left, and the photon shown
here carries sufficient momentum to reverse the direction of both
electrons. What we see in Figure 32-1 is a sudden, one-time change in the
paths of both electrons. Over a longer period of time, we would expect to
see many such interactions and on a larger scale, the path of each
electron would look more like a smooth curve. But since energy is thought
to be quantized, these interactions are fundamentally discontinuous. There
is also a near-symmetry in this diagram that suggests some ambiguity in
the process. Is it possible that the electron on the <em>left</em> emits
the photon, which is absorbed a few microseconds previously by the one on
the <em>right</em>? Regarding Feynman diagrams, Richard Halpern writes:<!--p .68-->
"In such pictures, backward signals seemed just as logical as forward
signals. Feynman saw no need to worry or philosophize about the lack of
causality. In his view, nothing in the Einsteinian universe dictated that
cause must always precede effect." Or perhaps there is no photon at all,
as long as the two electrons lie on one another's light cones:
<!--p. 56-->"What if there were no electromagnetic fields and just a pure
causal connection between the electrons?" Such a direct interaction, going
in both directions of time, has been called a <dfn>Wheeler-Feynman
handshake</dfn>. Or, as James Gleick put it when writing about Feynman's
theories during this same period, what if<!--113--> "That's what light is:
interaction between electrons[?]"</p>
<p>You see, the problem with fields is that their magnitude blows up to
infinity at the presumed infinitesimal radius of the particles responsible
for the character of the field. How do you keep an electron from
interacting with the field at its own location, or acting upon itself? As
a graduate student at Princeton, Feynman had the idea of eliminating the
field to solve this problem; so in 1941 he and his advisor John Wheeler
took the idea to Einstein for his comments.
<!--Gleick, Time Travel. 111-112-->Einstein commented that this line of
thinking, though somewhat paradoxical, wasn't completely foreign to him.</p>
<p>What is not made explicit in this discussion so far (nor in any other
discussion that I know of) is that there is a frame of reference in which
the electrons actually come into contact, and in which the emission and
absorption events shown in Figure 32-1 are one and the same. That specific
frame of reference is that of the supposed photon being exchanged, and in
such a frame there is no separation in space or time across that null
interval. If there is no distance between the electrons in such a
reference frame, there is no need for an electromagnetic field to exchange
the energy. The field and the waves in it are just what the interaction
looks like from our sub-luminal standpoint.</p>
<p>We might ask why electron two on the right emitted the photon when it
did. Was that photon <q>addressed</q> to electron one,
or <q>to whom it may concern</q>? If the former, how could this be possible? If the latter, how
would the energy loss associated with arbitrary emission of photons be
sustainable over any period of time? Was the energy sent by electron two a
wave that went out in all directions, collapsing to a particle upon
absorption, or was it a particle with only a single direction all along?
All of these questions are mooted in the reference frame where the
emission and absorption events are identical.</p>
<p>For all other reference frames, we are constrained to address a
separation in space, and also in time. Recognizing that this separation is
relative rather than absolute, we might forgive ourselves for saying
(perhaps using quotation marks to emphasize the subjectivity of it all)
that one aspect of the interaction occurs <q>backward in time.</q> John Cramer,
a physicist at the University of Washington, proposed an interpretation of
quantum mechanics in 1986 that offers a more sensible alternative to the
Copenhagen interpretation and the Many-Worlds hypothesis which I hadn't
planned on dignifying with any mention here (in short, the many-worlds
interpretation supposes that all possible results of every quantum
measurement do in fact become realized, each in their own branch of
reality). Cramer's idea, called the Transactional Interpretation of QM (<a
href="https://en.wikipedia.org/wiki/Transactional_interpretation" target="_blank">TIQM</a>),
elaborates on the Wheeler-Feynman handshake concept; it posits that
quantum interactions are mediated by complementary sets of waves, one of
which goes forward in time and the other backward. One is the familiar
wave function discussed in Chapter Twenty-Seven, and the other is its
complex conjugate. When these waves cross one other in opposite directions
from one particle to another but in proper phase with one another, a
transaction can occur such as the exchange of energy and momentum. TIQM
remains a relatively obscure idea, not often mentioned in popular books.</p>
<p>So many mysteries of QM seem to become trivial when one considers the
spacetime interval and embraces both a broader sense of locality and a
more critical view of sequentiality. Find any popular text which
emphasizes the apparent impossibilities of entanglement and see what
remains after you cross out any language that implies that entanglement
and measurement happen one after the other and in different places. If the
spacetime interval is zero, then those separations are not absolute.</p>
<p>Having said that, now I have to admit that I have greatly oversimplified.
Entanglement also occurs over timelike intervals, and can involve
particles having rest mass that move at less than the speed of light.
Electrons are the most obvious example of massive particles demonstrating
entanglement. Do we need a <q>backward-in-time</q> interpretation such as TIQM
to explain those cases? Perhaps. But I am not quite ready to concede that
point until I am convinced that electrons are not a special case of
electromagnetic waves. I will elaborate on that idea in this book's
concluding chapter, where I will also write more about locality.</p>
<p></p>
<h4>The topology of the immediate</h4>
<p>There's a word that doesn't come up often enough in discussions of
relativity and quantum mechanics, and that word is <dfn>topology</dfn>.
Topology is a branch of mathematics that no mathematician seems to be able
to explain clearly; but for the rest of us, topology concerns links
between elements (or <q>nodes</q>) in a network. Having worked as an IT
professional in the telecommunications industry, the examples that come
most readily to my mind are computer networks and telephone networks. But
the same idea applies to social networks. You have friendship connections
with other individuals. These individuals in turn have their own
connections, some to each other and some to people with whom you have no
direct connection. Another related mathematical concept is <dfn>graph
theory</dfn>, which uses different terminology but essentially comes down
to connections/links between pairs of elements/nodes. In this context, a <dfn>graph
</dfn> is a network diagram.</p>
<div class="figdiv nofloat_600 float_300_66 float_1050_50 float_1200_33" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/6n-graf.svg.png" style="max-height: 180px;">
<figcaption>
<b class="figlabel">Figure 32-2</b>.
<span class="ficaption_description">
A simple graph of six networked elements. Source: <a href="https://commons.wikimedia.org/wiki/File:6n-graf.svg">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
</div>
<p>A graph may be <em>undirected</em>, as in Figure 32-2, with the links
between nodes having no specific direction, or <em>directed</em>. In a
directed graph, the relationship between elements may be asymmetrical, as
in a social media network in which one account may <q>follow</q> another
without being followed in return.</p>
<p>Where I am going with all of this is that the spacetime interval might be
considered the <em>topology</em> of spacetime. It defines the
connections between events. Events having a null interval between them
(events which each lie in the light cone of the other) are directly
connected and can be related causally, although not in an entirely
symmetrical manner. We know that such events can come to agreement in a
transactional way that seems to be unbiased about the relative direction
of time in most frames of reference; but information seems able to flow
only in one direction, being the direction of past to future in any frame
of reference where the two events are not simultaneous. In this sense,
spacetime would be represented by a <em>directed</em> graph.</p>
<p>Let's consider a typical entanglement scenario again with Figure 32-2 as
a reference. Two entangled photons having the same polarization are
emitted at event number three. Alice and Zeno measure the polarization of
the photon at events two and four. As lightlike paths, the intervals from
three to two and from three to four are null, or zero, as indicated by the
lines between the events. The interval between events two and four is <em>not
</em> null, but spacelike; there is no way for a light signal from two to
reach four, or vice versa. Alice cannot tell Zeno the result of her
measurement before Zeno must make his, and vice versa. And this is more or
less the same impossibility that is crucial to the EPR <q>paradox</q> and all
the thousands of pages of breathless writing about entanglement and
nonlocality.</p>
<p>What is usually ignored in such discussions is that, having spacelike
separation, events two and four have no strict time order; the standard
language is that Alice's measurement at event two <q>immediately</q> determines
the outcome of the measurement at a far-away event four, whatever that
means. What is <em>always</em> ignored in such discussions is that
events two and four both share an <em>immediate</em> connection with
event three. And the reason these connections are ignored is that from any
mortal frame of reference, these connections are in both directions of
time, and this separation in time has apparent consequences; Alice and
Zeno are both free to change the orientation of their polarizing filters
<q>after</q> the photons have been emitted
at event three but <q>before</q> they
perform their measurements. We are reluctant to accept that their
spontaneous decisions can be either anticipated by, or communicated
backward in time to, event three. It is only for the photons that the
events in question are simultaneous.</p>
<p>For at least one other writer, entanglement is intimately connected with
spacetime. In <q>The Abdication of Spacetime,</q> Donald D. Hoffman writes:</p>
<blockquote>
Brian Swingle and Mark Van Raamsdonk found that curved spacetimes
obeying Einstein's general theory of relativity can emerge from tensor
networks of entangled quantum bits. In this scenario, the insouciance of
entanglement is feigned. Entanglement itself is somehow the fabric that
holds spacetime together.<!--iPhone photos 18-Jan-2023 Know This p. 66 -->
</blockquote>
<p>Lee Smolin says something similar, but coming at it from the other
direction:
<!--TR p. 187-->"space arises from turning off connections in a network."
If the interval between <em>all</em> pairs of events were zero, then
there would be no space, no time.
<!-- Ball p. 93-94. The mystery of Wheeler's delayed-choice experiment [but! it disappears when the interval is taken into account] also appears in Halpern, 241-43 --></p>
<p>Concerning the mystery of this chapter's opening paragraph, why an idea
that appeals so much to me and seems so obvious should remain unmentioned
in popular literature, I may have found a clue in George Musser's <em>Spooky
Action at a Distance</em> as I have been working on this book's second
draft. Of Roger Penrose's <q>twistor</q> theory, Musser writes:<!--p. 205--></p>
<blockquote>
He initially tried to conceive of space as a network … he built his
new theory not on particles or other localized building blocks, but on
light rays. Penrose wasn't interested in light <i>per se</i> — as a means of
illumination — but in the causal links that light rays represent. Light
rays stretch infinitely far across space, so they're about as nonlocal
as you can get.
</blockquote>
<p>It seems that the theory hasn't caught on because certain details haven't
been worked out. I'm very grateful for Musser's plain-English summary,
because Penrose's 75-page paper introducing the subject<!--https://www.sciencedirect.com/science/article/pii/0370157373900082?via%3Dihub--> ("Twistor
theory: An approach to the quantisation of fields and space-time,"
co-authored in 1973 with M. MacCallum) is at least as far over my head as
the treatment he gives it in the penultimate chapter of <cite>The Road to
Reality</cite>,<!--p. 962 and following--> which he wrote with the intent
of reaching a more general audience in 2005. Musser writes that physicists
inspired by the likes of Penrose and John Wheeler have been working since
the 1960s to make similar approaches work, in which the network is the
fundamental thing from which space emerges. Fotini Markopoulou is among
those leading that effort today.</p>
<h4>Why things fall down, at last</h4>
<p>In Chapter Twenty-Five, we considered the question of why things fall
down in the theory of relativity, and we almost got there without having
to invoke the curvature of <q>time.</q> My argument was that the curvature of
space was sufficient to explain the direction of acceleration for a body
already in motion relative to the earth<!--or did I say it was relative to the observer?-->,
but not for explaining why something would drop toward the earth if
released from a state of rest above it. Loath as I am to invoke time as a
fourth dimension and the curvature of time as a cause of gravity, I will
propose here a quantum-mechanical alternative or equivalent explanation.</p>
<p>For one last time, we will consider Zeno's lump of mashed potatoes from
Chapter Six. Zeno holds the lump six feet above the surface of the flatbed
truck on which he stands, and then lets go of it. Why does it drop? We
want an answer that relies strictly on the nature of the lump and on the
curvature of space in its immediate vicinity, not the relative motion of
any observer nor on any <q>principle of maximal proper time</q> that
presupposes when and where the lump must hit the truck bed and then offers
to tell us the path it takes to get there.</p>
<p>I think the real explanation of why the lump falls down is that it all of
its parts are in a constant state of indecision regarding where they
really are or want to be. The electric forces and chemical bonds between
their component molecules do tend to keep them together, though. If enough
of them decide to go in a particular direction, the lump as a whole will
tend to move in that direction. The same can be said of the particles
making up their atoms. If the electrons decide to fall, the nucleus will
follow. If we can explain the fall of the electrons, our explanation is
complete.</p>
<p>Heisenberg tells us that the electrons don't even <em>have</em> a
well-defined position, at least some of the time. But if they were made to
pick one, the odds are slightly in favor of a downward direction. Because
of the curvature of space, there are just more places to be down than
there are to be up. And if we consider the electrons as de Broglie did, as
waves rather than particles, then the electrons are always in oscillatory
motion relative to the earth, and the curvature of space must refract them
downward. I'll have a little more to say about the wave nature of matter
in this book's conclusion.</p>
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<hr>
</div><div id="Fit8">
<h3 class="fit" id="HallOfMirrors">Fit the Eighth<br>
<i>The Hall of Mirrors</i></h3>
<p>Alice Lary had returned from her holiday in Italy to the council flat in
Ladbroke Grove she shared with her twin cats named Albert and Niels and
her roommate Priya. She flipped through the pile of letters that had come
by post during her absence, pondering the fact that upon returning home,
bills and junk mail seldom fail to ground a person back in the unpleasant
realities of their day-to-day life. She made herself a cup of tea,
ignoring a student loan statement and tossing away junk mail. A telephone
call earlier that day had brought her the news that on Monday when she
returned to the call center where she worked, she would be taking on a new
role as supervisor (but without any increase in pay), replacing her boss
who had been gradually going mad over the course of the past year and had
finally consented to hospitalization. Alice sang to herself as she wound
up a ball of yarn that Niels had gotten all in tangles:</p>
<blockquote>
<em>
<p>"Clockwork and brickwork and balls dropped from towers<br>
Comets and planets and meteor showers <br>
Pendulum bobs and weights dangling from strings<br>
Would you believe that they all are just strings?</p>
<p>The particle zoo down to every last species<br>
Tenures and fellowships, doctoral theses<br>
Marionettes dancing and caged birds that sing<br>
Theorists say that they're all made from string</p>
<p>I often wish<br>
I had not quit<br>
As an undergrad<br>
But when I'm reminded of such useless —"</p>
</em>
</blockquote>
<p>Alice stopped singing to scold the cats, who were now fighting over a
ball. The ball was supposed to make rattling noises as it rolled, but had
somehow lost whatever contents had made it do so. She set her tea down
next to her chair and scooped both cats into her lap as she sat down.</p>
<p><q class="dialog">Why must you always quarrel so?</q> She asked them.
<q class="dialog">Why don't you sing me
your duet like Don Henley and Stevie Nicks?</q> As she stood Albert on his
hind legs to take the first turn at an imaginary microphone, Niels
scampered away. No duets today. She set Albert back down again and sipped
at her tea while stroking his fur. On the bookshelf next to her chair were
a number of plays by Tom Stoppard, all of which but <cite>Arcadia</cite>, his
latest, showed considerable wear from multiple readings. All five of
Douglas Adams' <cite>Hitchhiker's Guide to the Galaxy</cite> novels were
there, and the complete works of Shakespeare in a handsomely bound volume
Alice's mother had given her. Alice reflected on how fun her vacation had
been, relishing the ideas that had come to her at least as much as the
memories of people and food and shops and scenery, all written together in
her diary. She was looking forward to sharing them on usenet, and with
that she began to wonder what kind of replies had followed her last post,
a pseudonymous satire attributed to an imaginary conversation between
Lewis Carroll and Albert Einstein. The usenet and BBS posts were her
creative outlet and her cognitive process as she worked her way through
the mysteries of modern physics. As Alice returned to her longstanding
contemplation of the apparent dichotomy between relativity and quantum
mechanics, she fell asleep.</p>
<p>Alice found herself in a long corridor. On each wall were hung a number
of mirrors, spaced so that each mirror faced a counterpart on the opposite
wall. None of the mirrors' corners were square, and each had the
shape of a parallelogram.</p>
<div class="figdiv nofloat_600 float_300_66 float_1200_50">
<figure>
<img class="Figure" loading="lazy" src="images/hallofmirrors.png" style="max-height: 480px;">
</figure>
</div>
<p>Walking down the hall, Alice saw a picture on the wall showing what
appeared from a distance to be a Cartesian grid. On closer inspection it
was recognizable as a chessboard, but divided vaguely into quarters as if
by coordinate axes. Across the center line between the opposing sides, the
positions of each piece on the board were mirrored by an identical
opposing piece.</p>
<div class="figdiv nofloat_600 float_300_66 float_1050_50 float_1200_33">
<figure>
<img class="Figure" loading="lazy" src="images/chessboard.png" style="max-height: 280px;">
</figure>
</div>
<p>In the nearest mirror, Alice saw the image of an equation: "<i>z</i> =
<i>x</i> + <i>iy</i>." In that mirror's opposite, across the hall,
she saw <q><i>z</i>* = <i>x</i> - <i>iy</i>.</q> Alice came next to a
framed light cone diagram, and then another mirror in which she saw
countless reflections of herself shrinking into the mirrored distances
created with this mirror's counterpart behind her on the opposite wall.
She waved to herself, and each successive reflection waved back after a
barely perceptible delay due to the lag of the light crossing back and
forth between mirrors, one additional instant for each successive
reflection. <i>This is a mirror into the past,</i> Alice concluded. She
wondered momentarily if she were to turn around whether she might be able
see her future in the opposite mirror. But each of the paired mirrors
behaved in the same way. <i>To see into the past, I would probably
require eyes in the back of my head</i>, she thought. She thought of
the White Queen in <cite>Through the Looking-Glass</cite>, who remembered
things that hadn't yet happened.</p>
<blockquote>
<p></p>
</blockquote>
<p>Moving on, Alice found a lenticular picture of a cat in a box; depending
on the viewing angle, the cat stood or lay down.</p>
<div class="figdiv nofloat_600 float_300_66 float_1050_50 float_1200_33">
<figure>
<img class="Figure" loading="lazy" src="images/cats.gif" style="max-height:200px;border: 2px solid #555">
</figure>
</div>
<p>The final pair of looking-glasses were different somehow, not reflecting
quite as much light as the others. Looking past them, Alice saw that the
last picture on the wall was of a deck of cards scattered randomly on a
flat surface. Though only some of the cards were face-up, this being a
dream, Alice was able to instantly perceive that no two of the cards
shared the same suit or face value. Atop the pile were the king of
diamonds and the queen of hearts. Alice thought of Wonderland's Red Queen,
full of rage and caprice, so different from her counterpart on the
chessboard.</p>
<p>Coming square with the last pair of mirrors now, Alice realized that she
was able to see her two cats in them, but only one of them at a time as
she faced one mirror or the other. The glass was slotted somehow and only
permitted direct viewing, like a traffic light that can be seen only from
positions near the stop line. Or perhaps like a polarizing filter? She
paused for a moment, seeing herself in the mirror as if holding one of her
cats. In this position, Alice was blocking her own view and could not see
the other cat's position in the reflection of the opposite mirror. She
turned around to face the other mirror, and then saw herself holding her
other cat. <i>How can I know that both cats are there if I can only see
one at a time?</i> she asked herself, beginning to feel anxious. She
started to retrace her steps down the hall, toward the place where she had
come in.</p>
<p>Passing the picture of the boxed cat again, Alice came to the mirrors of
future and past, but this time she saw her cats in these mirrors as well,
and something was wrong. To all outward appearances, they were whole, but
something told Alice that they were both in mortal peril and that between
the two of them they had only one set of the vital organs required for
their survival. It was up to her to decide which cat she would sacrifice
for its parts and which cat would be made whole; which would belong to her
future and which would belong to her past. Alice was heartbroken over her
dilemma and awoke crying silently.</p>
<p>Through eyes blurred with sleep and tears, Alice saw the image of a stout
little man. As her vision became clearer, the image resolved into the a
portrait of Tweedledee and Tweedledum that hung on her wall. Seeing both
cats in front of her, still at play, she collected herself and realized
that the cats that she had seen in her dream as <em>two</em> must have
actually been <em>one and the same</em>; It had been Albert in both of
the paired mirrors of past and future, and it had been Niels in the facing
pair of mirrors next to Albert's .</p>
<p>Alice picked up a pen and began to write.</p>
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<hr>
</div><div id="Conclusion">
<h3 id="EinsteinsCat">Conclusion<br>
Einstein's Cat</h3>
<p>At long last, we return to the cat whose fate was left hanging in the
balance in this book's introduction. To recap, it is 11:57 AM in
Princeton, New Jersey where Henry Ford is chatting about relativity with
Albert Einstein. At 12:00 noon in Dearborn, Ford's cat is to die, and
there isn't anything he can do about it; he can't even say goodbye,
because by the time he gets to the nearest telephone it will be too late.
Continuing his lesson, Einstein asks Ford regarding these circumstances,
<q>Is there any strict sense in which your cat is not already dead?</q></p>
<p>At this point, I will admit the pretense of Ford ever having owned such a
cat, and of such a cat ever having been in any such danger. This is
because although I might be able to believe that Einstein was capable of
the insight (and perhaps even the malice) to construct this thought
experiment as I have related it so far, I do not believe that he ever
could have been party to such a scheme; but more importantly, I have no
reason to believe that Einstein himself would have paid particular
attention to the full implications of such a scenario, which implications
I will now relate.</p>
<p>In partial answer to Einstein's question, there is some sense in which
Ford's cat is not already dead, if we stipulate that it is 11:57 AM in
Dearborn just as it is 11:57 AM in Princeton. Ford's cat can still do
things that Ford will be able to have knowledge of later. It can meow and
scratch at the walls of the box in which it is trapped. If it knew how, it
could leave a goodbye message that Ford could read. If all is as Einstein
described it, the cat can and will die, and Ford can bear personal witness
to this fact at some time in the future when he and the cat are once again
in the same place. Ford can hear the sad news second-hand as soon as he is
able to get to a telephone.</p>
<p>So in that sense, the cat is still alive. On the other hand, there is
still the sense in which the cat is already dead, because Ford will never
see it alive again, and the cat will never hear from him again. So we
might say the cat is only <em>half</em> dead, or in an indeterminate
state of being both alive and dead, like its more famous counterpart in
Schrödinger's thought experiment. But there is no <em>strict</em> sense
in which the cat is either wholly alive or dead, because it is not
entirely meaningful to ask what time it is (now) in some other place. Or
perhaps more precisely, the further away some place is from <em>here</em>,
the more obviously ambiguous the answer is to the question of what time it
is there, <q>now.</q> The distance between Dearborn and Princeton is somewhat
trivial in relation to the speed of light, and modern advances in
telecommunications greatly reduce the ambiguity of time between these two
places from what it would have been in 1935; Henry Ford never owned a
cellular phone, and the story of his cat as I presented it depends on that
fact.</p>
<p>So from a modern perspective, let's look at the problem over a greater
distance. Suppose Alice is in London and that Zeno is on a moon base
250,000 miles away, and the two of them are having a live video
conference. Behind each of them is a clock that the other can see. If it
is exactly 12:00 noon in London, Alice could suppose it is 11:59:58.72 at
Zeno's moon base, because that would be the time she sees on Zeno's clock.
On the other hand, Alice might conclude in consultation with Zeno that it
is 12:00:01.28 on the moon, because that is the time Zeno's clock will
read when he sees Alice's clock reach exactly 12:00 noon. The truth that
encompasses both of those perspectives is that whatever happens either in
London or on the moon base in the 1.28 seconds it takes for light to
travel in-between them has nothing at all to do with <q>now</q> at either
place.</p>
<p>As we conceive things currently, 1.28 seconds of video is said to be
stored as waves in the electromagnetic field between Alice and Zeno,
inaccessible by either one of them until its time has come and its journey
is complete. But if we don't intercept any of that energy along the way,
if we don't <em>measure</em> it, then it might just as rightly be said
to vanish at the event where and when it is transmitted and to reappear at
the event where and when it is received; and in Chapter Thirty-Two, we saw
that on a moment-by-moment basis there is a frame of reference in which
those two events are for all intents and purposes one and the same.</p>
<h4>Hidden variables</h4>
<p>You may happen to read that the experiments which proved John Bell
correct have eliminated the possibility of the <q>hidden variables</q> Einstein
sought to explain entanglement in a deterministic way (Chapter
Twenty-Seven). This isn't quite correct. I hope that I have made clear
that what we must give up on is our primitive concept of locality. Bell's
theorem grew from Bell's efforts to evaluate an interpretation of quantum
mechanics offered in 1952 by David Bohm (not to confuse Bohm with Bohr or
Born may be difficult, I know). Bohm supposed that electrons were
particles having one and only one location, but that there was a wave
associated with them that scouted out the paths ahead and determined which
of them the electron would take. This was called a <q>pilot wave,</q> an idea
proposed by de Broglie<!--https://web.archive.org/web/20110819234038/http://leopard.physics.ucdavis.edu/rts/p298/pilotwavetheory.pdf-->.
Bell favored the theory and wrote in 1982:</p>
<blockquote>
Bohm showed explicitly how parameters could indeed be introduced, into
nonrelativistic wave mechanics, with the help of which the
indeterministic description could be transformed into a deterministic
one. More importantly, in my opinion, the subjectivity of the orthodox
version, the necessary reference to the <q>observer</q>, could be eliminated.
… But why then had [Max] Born not told me of this <q>pilot wave</q>? If
only to point out what was wrong with it? Why did von Neumann not
consider it? More extraordinarily, why did people go on producing
<q>impossibility</q> proofs, after 1952, and as recently as 1978? … Why is
the pilot wave picture ignored in text books? Should it not be taught,
not as the only way, but as an antidote to the prevailing complacency?
To show us that vagueness, subjectivity, and indeterminism, are not
forced on us by experimental facts, but by deliberate theoretical
choice?
</blockquote>
<p>In this essay, Bell is essentially saying that "Something is rotten in
the state of Denmark" and that determinism didn't die a natural death in
Copenhagen, but it was in fact murdered and still comes back to haunt us
like Hamlet's slain father. <i>Der Kopenhagener Geist</i>, indeed. He
writes that even though (in the tradition of John von Neumann) there were
still physicists declaring the <q>impossibility</q> of hidden variables and of
determinism, Bohm's deterministic interpretation was valid in that it was
nonlocal. It is <em>local</em> determinism that Bell rejected. In <cite>Something
Deeply Hidden</cite>,
<!--p 190-->Sean Carroll writes: "Bohmian mechanics is perfectly
deterministic, but it is resolutely nonlocal. Separated particles can
affect each other instantaneously [sic]." And furthermore:</p>
<blockquote>
The Bohmian world is completely quantum, not stooping to an artificial
split between classical and quantum realms. … Bohmian mechanics is an
explicit construction that does what many physicists thought was
impossible: to construct a precise, deterministic theory that reproduces
all of the predictions of textbook quantum mechanics, without requiring
any mysterious incantations about the measurement process or a
distinction between quantum and classical realms. The price we pay is
explicit nonlocality in the dynamics.
<!--193-94-->
</blockquote>
<p>Lee Smolin shares Bell's enthusiasm for Bohmian mechanics and his dismay
regarding the dominance of other, less sensible, interpretations of QM. He
calls it a <q>scandal</q> that Bohm's pilot wave theory is all but kept secret
from physics students. In <cite>Time Reborn</cite>, Smolin considers the
problem of hidden variables at some length and asks
<!--p. 155-->whether any subsystem of the physical world can be given full
consideration in isolation from its larger context.</p>
<p>The answer to that seems to be a resounding <q>no.</q> What physicists call a
<q>hidden variables</q> problem, mathematicians and computer programmers might
call an <q>initial value problem.</q> Both QM and relativity are founded on
differential equations (Chapter Fourteen) and thus require a set of
initial values that can be carried either forward or backward in time. One
requires a fully-specified starting point. But that's just the thing about
QM: there is no such full specification. The precision of some quantities
comes only at the expense of the precision of others. What is far more
seldom talked about is that relativity has the same dilemma, not in the
Heisenberg uncertainty principle <i>per se</i>, but in mutual
interactions being limited by the speed of light.</p>
<p>I once tried to model gravitational interaction with a computer program.
On a Cartesian grid, I planned to set up two imaginary, freely falling
bodies with known initial conditions for mass, position, and velocity; and
then see how they interacted gravitationally. Having at the time no grasp
at all on general relativity, I instead chose the classical force law for
gravity given by Newton. But I wanted to at least be sophisticated
enough to use the speed of light as a limiting condition on how the bodies
interacted. Rather than implementing gravity as an instantaneous <q>action
at a distance</q> causing each of the bodies to be attracted to the position
the other <i>is</i>, I decided that they would be attracted to the
position the other <i>had been in</i>. In other words, information about
each body's mass, position, and velocity would be carried to the other at
the speed of light, putting my <q>experiment</q> at least partially in
accordance with the theory of relativity.<!--Wheeler p. 187 and following mentions the delayed time in gravitational interaction --></p>
<p>After some consideration, I decided that the problem need not be a hard
one. To determine the immediate force on Body A, I only needed to figure
out which event in its past light cone (Chapter Seven) was occupied by
Body B. Once I had that event, I could take the spatial distance between
that and the event corresponding to Body A's immediate event, plug it into
Newton's gravitational force law, and calculate the force. To
determine the immediate force on Body B, I would do the same calculations
with regard to Body A. This algorithm was to be repeated over and over at
intervals of time to create an approximation of the paths that the bodies
would follow if they were acted on continuously. For each iteration, I
would increase the time by a value of one <q>tick</q> of my imaginary clock.</p>
<p>Once I tried to implement this solution, however, several problems became
immediately obvious. Starting only with the initial conditions (at
time zero), I had no idea where Body B intersected Body A's past light
cone. Since the two would already have been interacting gravitationally,
putting them on curved paths, I could only extrapolate linearly backward
from Body B's zero-time position and velocity, and make it an explicit
condition of the test that both A and B had been released from uniform
motion at time zero. For each successive calculation of the immediate
force on Body A, my program would have to ask first whether Body B was
close enough and whether enough time had yet passed for Body A to be
affected by B's having been released from rest at time zero. If not, my
program could safely assume that B had been on a linear path matching its
position and velocity at time zero and then make the force calculations
accordingly.</p>
<p>The second problem concerned how to track all of the changes in velocity
that the bodies experienced after being released from rest. It wasn't
enough just to track their current positions and velocities; At each
iteration of my algorithm — for each tick of the clock — I had to be able
to recall the position and velocity of Body B at the time it had crossed
the past light cone of the event which Body A currently occupied. I
wondered whether each point in my imaginary space could be made to receive
information about the motions of these bodies and propagate them through
the grid at the speed of light. I imagined that for each tick of the
clock, I could take the information at one point and pass it to its
nearest four neighbors. This however, presented its own problem:
light propagates radially, spreading out in a circle, while nodes in a
square grid can only interact in an inherently square fashion. I decided
that my approach would be to store these positions and velocities in a
database table during each iteration, starting from time zero, so that
they could be recalled when needed.</p>
<p>The third problem was that I would be storing my data at integer values
of time, yet I would almost certainly be looking for position and velocity
measurements which had occurred at non-integer values of time. In other
words, I could expect Body B to have intersected Body A's light cone at
times <em>in-between</em> ticks of the clock rather than always <em>on</em>
them. I would have to estimate using the nearest neighboring measurements.</p>
<p>I assumed that real physicists must have a better approach to these types
of problems and I wondered what it was. Shortly later I became distracted
by other activities. What I failed to appreciate at the time I was writing
this program was that the problems I encountered were themselves
indicative of the true nature of our world. The first problem was by far
the most interesting. I had assumed that the precise positions and
velocities of both masses were adequate to fully specify the initial
conditions of the program. But that still wasn't enough information to run
the program forward, letting the laws of physics take over. For that
matter, it wasn't enough to run the program <q>backward</q> either. What was
still missing was the information about the <q>past,</q> which in some
relativistic frames of reference was simultaneous to certain events at my
time zero. The information about which precise force gravity was supposed
to exert on each body was <q>stored</q> in the gravitational field, just like
the 1.28 seconds of video traveling between London and the moon base, or
the sound of the chiming clocks in Chapter Seven and the <q>Circus Canon.</q>
(Note: I recently asked an <em>actual</em> computational physicist how he
or his colleagues might handle the calculations for such a simulation. To
my great surprise, his answer was more or less that "we don't really think
about that.")</p>
<p>My attempt at a computerized <q>thought experiment</q> is what Smolin calls
doing <q>physics in a box.</q> The inherent fallacy in such things is that they
can only succeed in a classical physics paradigm in which initial
conditions can be fully specified with arbitrary precision. Such a
paradigm results in <q>The Expulsion of Novelty and Surprise</q> (the title of
Smolin's fifth chapter in <cite>Time Reborn</cite>). What gives us a universe
of guaranteed surprises is that the interactions between any two things in
it are without beginning or end. They are, in quantum mechanical terms,
inseparable.<!--III:46 holography--> Non-locality and separability
(Chapter Twenty-Seven) are the key things at issue in the EPR <q>paradox</q>
and in Bell's theorem.</p>
<p>Let's imagine a <q>box</q> to do physics in, a bounded volume of space over a
limited period of time from zero to <i>t</i>. A <q>span</q> of spacetime, if
you will. The <q>initial conditions</q> in that span of spacetime are that
everywhere in it there are force-carrying waves or particles that
originated outside that boundary or <q>span</q> and have yet to accomplish
their mission. There are light waves and gravity waves that carry
information about things that happened prior to time zero and/or outside
that bounded space.
<!--there's that <q>boundary</q> again-->These
are the <q>hidden variables</q> that
make predictions impossible beyond a certain degree of precision.</p>
<p><q>Hidden variables</q> are exactly what Einstein found missing from quantum
theory, leaving us a world in which some processes are supposed to happen
completely at random, in a sense without cause. Quantum theory demands
indeterminacy, while relativity is seen by some (including Smolin) as
giving us a four-dimensional, pre-determined <q>eternity.</q> Relativity, in
the traditional view, is supposed to <em>codify</em> causality by
setting a speed limit on any and all processes, while quantum theory
appears to <em>defy</em> causality. Smolin sees a complete absence of
indeterminacy in relativity, denouncing the Newtonian <q>Expulsion of
Novelty and Surprise</q> presenting relativity as the final nail in the
coffin of free will: <q>Given the initial conditions, Einstein's equations
determine the whole future geometry of a particular spacetime and
everything it contains, including matter and radiation,</q> leaving <q>no role
for or sign of our awareness of the present.</q>
<!--(Time Reborn, p. 71)-->Ironically, in the text preceding this, he
seems to demonstrate precisely the opposite, showing in a very salient
analysis that the <q>initial conditions</q> of a <q>subsystem</q> are necessarily at
least partly <em>outside</em> the subsystem being considered: <q>We live
in a world in which the flap of a butterfly's wing can influence the
weather oceans away and months later.</q> <cite>Time Reborn</cite> is one of my
favorite books on physics and one of the most frustrating books I have
ever read, both precisely because Smolin gets <em>so very close</em> to
seeing what I see.</p>
<p>Anyone having a passing familiarity with quantum physics knows that there
is a degree of uncertainty inherent in the measurement of position and
momentum of any mass, as made evident by Werner Heisenberg (Chapter
Twenty-Seven). This uncertainty is most evident on a small scale, and this
small scale is the usual context in which quantum physics is considered.
The nature of this uncertainty is that the more accurately we measure a
particle's position, the less we can know about its momentum, and vice
versa. Mathematically, we have the standard deviation of the position <i>x</i>,
multiplied by the standard deviation of the momentum <i>p</i>, always
being greater than or equal to a universal constant:</p>
<blockquote class="equation">
<p>σ<i><sub>x</sub></i>σ<i><sub>p</sub></i> ≥ <i>ħ</i>/2</p>
<b>Equation 33-1.</b> <span class="equation_description">The uncertainty relation between position
and momentum.</span>
</blockquote>
<p>Rather than as virtual objects in a computer program, let us re-imagine
bodies A and B as enormous but quantified masses widely separated in
space, perhaps at a distance of ten light-seconds (by comparison the
distance between the earth and sun is around eight <q>light-minutes</q>). At
time zero, we measure their positions and velocities (how you would do
this with such large and widely-separated objects is beyond me, but for
the sake of argument let us assume that we can). Knowing their mass, we
can calculate their momentum from their measured velocity. With what
degree of certainty can we predict what their positions and momenta will
be one second later? Those quantities will be affected by their mutual
gravitational attraction during that second. With what degree of precision
can we calculate that attraction? We have their positions now, but how
could these bodies react to each other's present locations or actions if
they are ten light-seconds apart? Mustn't they each instead be attracted
to the forces that result from their counterpart's position ten seconds
ago? Given that they have been interacting long before we arrived on the
scene to take measurements, we can only make an educated guess as to their
past positions, and thus there is an amount of uncertainty regarding the
path they will follow and what the position and momentum of each will be
one second from now. As we take repeated measurements, this uncertainty
decreases. For every second that passes, we have that much less history of
interaction to speculate about and can instead refer to our previous
measurements.<!--"What these equations tell us about the relationship between position and frequency (or wavenumber) is that they are (for want of a better term) holistic."--></p>
<p>A less-talked-about consequence of Heisenberg's uncertainty principle is
that measurements of energy decrease in uncertainty over time. If we
take liberties with the above uncertainty equation, we may grope toward a
sense of why this may be so: if we multiply position with momentum, we get
the same units of measure as energy multiplied by time. As Lawrence Krauss
explains, <q>[T]he uncertainty principle tells us that the longer we measure
something, the more accurately we can determine its total energy. Since
all measurements take merely a finite time, however, there is always a
residual uncertainty in the value of the energy that can be measured in
any system.</q>
<!--(Hiding in the Mirror, p. 108)(III-81)--></p>
<p>This sounds suspiciously like our computer modeling problem; at time
zero, we have no idea what gravitational energy is hidden in the system.
It would be revealed only over time as gravity waves (or gravitons)
carrying energy in the gravitational field reach their targets. If
we had instead chosen to model an electrodynamic system consisting of two
charged particles, the problem would have been the more or less the same
as our gravitational system, with some of the information in the system
being tied up in photons rather than gravitons. In either case, the
force-carrying particles contain <q>hidden variables.</q> Is it possible that
relativity and quantum theory are thus different expressions of the same
physical limits?<!--limits of information …--></p>
<p>Cramer's Transactional Interpretation (Chapter Thirty-Two) frames quantum
interactions as complex-conjugate wave functions, one going forward in
time and the other backward. The standard visualization of relativity's
spacetime is a pair of cones pointing in opposite directions of time, the
shape of each determined by a spherical wavefront of light.</p>
<h4>How to build a light clock</h4>
<p>Though I can't find the phrase again by searching for it on line, I was
sure that I read somewhere that Werner Heisenberg once said that physics
is an honest profession, and anyone wishing to comment on it is obligated
to learn it first. I'm going to be talking out of turn quite a lot in this
last part of the book, lifting words and ideas from contexts I don't fully
understand and putting them together in a way that may make sense only to
readers whose education is as lacking as mine. However enlightening this
presentation may <em>seem</em>, bear in mind that this is only "the way
things appear from the perspective of one curious amateur" and is not the
considered opinion of any single expert. John Gribbin wrote:<!--p. 123-->
"Quantum theory does not say what atoms are like, or what they are doing
when we aren't looking at them." Since I don't really know better, I'm
going to attempt to do that here for your entertainment.</p>
<p>In <cite>Through the Looking-Glass</cite>, Alice finds herself on a
chessboard, running with the Red Queen, but somehow staying in one place.</p>
<div class="figdiv nofloat_600 float_300_66 float_1050_50 float_1200_33" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/1599px-Alice_queen2.jpg" style="max-height: 280px;">
<figcaption>
<b class="figlabel">Figure 33-1</b>.
<span class="ficaption_description">
Alice and the Red Queen race across the chessboard.
</span>
</figcaption>
</figure>
</div>
<blockquote>
<p><q class="dialog">Well, in our country,</q> said Alice, still panting a little, <q class="dialog">you'd