Updated
with numerous links!
Are
Perceptual Fields Quantum Fields? by Brian J.
Flanagan
Key Words: Perceptual
Fields, Quantum
Fields, Vision, Space, Mind-Brain Identity, Hard Problem "Out
of the multitude of our sense experiences we take, mentally and
arbitrarily, certain repeatedly occurring complexes of sense
impression... we attribute to them a meaning — the meaning of the
bodily object. Considered logically this concept is not identical with
the totality of sense impressions referred to; but it is an arbitrary
creation of the human (or animal) mind. On the other hand, the concept
owes its meaning and its justification exclusively to the totality of
the sense impressions which we associate with it" (Albert Einstein) Acknowledgements
1)
Introduction In
a widely
quoted Scientific American article
of recent vintage, David
Chalmers
suggests we look to information states as a possible means of bridging
mind and matter: The
abstract
notion of information, as put forward by Claude
E.
Shannon
of MIT, is that a of a set of separate states with a basic structure of
similarities and differences between them. We can think of a 10-bit
binary code as an information state, for example. Such information can
be embodied in the physical world. This happens whenever they
correspond to physical states (voltages, say); the differences between
them can be transmitted along some pathway, such as a telephone line. We
can also find information embodied in conscious experience. The pattern
of color patches in a visual field, for example, can be seen as
analogous to that of pixels covering a display screen. Intriguingly, it
turns out that we find the same information states embodied in
conscious experience and in underlying physical processes in the brain.
The three-dimensional encoding of color spaces, for example, suggests
that the information state in a color experience correspond directly to
an information state in the brain. We might even regard the two states
as distinct aspects of a single information state, which is
simultaneously embodied in both physical processing and conscious
experience. [1] Chalmers
is
best known for his formulation of the “hard problem”, i.e., how
to explain the brain’s phenomenal states? Let us compare Chalmers’
views with a proposed solution to the mind/body problem from Herbert
Feigl,
one of
the most eloquent proponents of mind/brain identity theory: The
solution that appears most plausible to me, and that is consistent with
a thoroughgoing naturalism, is an identity theory of the mental and the
physical, as follows: Certain neuro-physiological terms denote (refer
to) the very same events that are also denoted (referred to) by certain
phenomenal terms. The identification of the objects of this twofold
reference is of course logically contingent, although it constitutes a
very fundamental feature of our world as we have come to conceive it in
the modern scientific outlook. Using Frege's distinction between Sinn
('meaning', 'sense', 'intension'), and Bedeutung ('referent',
'denotatum', 'extension'), we may say that neurophysiological terms and
the corresponding phenomenal terms, though widely differing in sense
... do have identical referents. I take these referents to be the
immediately experienced qualities, or their configurations in the
various phenomenal fields. [2] On
the present view, the key to the reconciliation of mind and matter is
contained in embryo in Chalmers' remarks, specifically: "the
three-dimensional encoding of color spaces, for example, suggests that
the information state in a color experience corresponds directly to an
information state in the brain. We might even regard the two states as
distinct aspects of a single information state, which is simultaneously
embodied in both physical processing and conscious experience". And now
Feigl again: "we may say that neurophysiological terms and the
corresponding phenomenal terms, though widely differing in sense ... do
have identical referents. I take these referents to be the immediately
experienced qualities, or their configurations in the various
phenomenal fields". It
is worth
noting that the views expressed above are consonant with those found in
Bohm
and Hiley: "one may then ask what is the relationship between the
physical and the mental processes? The answer that we propose here is
that there are not two processes. Rather, it is being suggested that
both are essentially the same" [3] We
consider that one might well consider Chalmers’ “two” information
states as complementary aspects of a single state. More particularly,
we ask whether one might regard the states of perceptual fields and
their concomitant quantum fields as being somehow the same. Feigl’s
“immediately experienced qualities, or their configurations in the
various phenomenal fields” might then be identified with ‘observables,
or their configurations, in various quantum fields’ - what amounts to a
mind/brain identity theory wherein the matter of the brain is viewed at
the quantum level. Intriguingly,
the mathematics of quantum mechanics and quantum field theory offers a
number of striking parallels with the mathematics of color and the
visual field. We shall have a look at a handful of such parallels, in
respect of a few elementary aspects of vectors, manifolds, fields,
symmetry, logic, and group theory. Given the broad interest in such
matters today, we will light our way with the assistance of a number of
introductory texts drawn from a dazzling array of luminaries past and
present. 2)
Apology How
might quantum theory help us with neuroscience? At the present
historical juncture there are not a few who would readily reply ‘not at
all’. And indeed the polemical storms concerning the issues at hand
have been something to behold these last few years. By way of reply, we
refer first to Bohr,
the father of quantum mechanics (QM) who, as Bohm tells us, “suggests
that thought involves such small amounts of energy that
quantum-theoretical limitations play an essential role in determining
its character.” [4] Next,
we
bring in Freeman Dyson,
who states quite clearly that, from the perspective of quantum field
theory (QFT)
everything in the physical universe just is a quantum field: “There is
nothing else except these fields: the whole of the material universe is
built of them.” [5] Thus, the brain just is a collection of fields,
which is interesting, given that the objects of our immediate awareness
are also fields, viz., perceptual fields. We
consider
that the most relevant physical fields must be electromagnetic (EM)
fields, and in this wise we quote Abdus Salam,
who reminds us that “all chemical binding is electromagnetic in origin,
and so are all phenomena of nerve impulses.” [6] We reason that, if
perceptual fields are “phenomena of nerve impulses”, as would seem
altogether plausible, then it would seem to follow that perceptual
fields are “electromagnetic in origin”. Further
support for our thesis comes from Karl Pribram,
who, in a classic modern work, writes: “The text of this volume claims
that the mathematical formulations that have been developed for quantum
mechanics and quantum field theory can go a long way toward describing
neural processes due to the functional organization of the cerebral
cortex.” [7] We
want to go a little further than Pribram and regard the brain and sense
organs as thoroughly quantum, through and through. So saying, we
produce a passage from Umezawa’s
highly accessible work on Advanced Field
Theory: Among
the many biological objects a particularly interesting one is the
brain. For any theory to be able to claim itself as a brain theory, it
should be able to explain the origin of such fascinating properties as
the mechanism for creation and recollection of memories and
consciousness. For many years it was believed that brain function is
controlled solely by the classical neuron system which provides the
pathway for neural impulses. This is frequently called the neuron
doctrine. The most essential one among many facts is the nonlocality of
memory function discovered by Pribram (...) There
have been many models based on quantum theories, but many of them are
rather philosophically oriented. The article by Burns...provides a
detailed list of papers on the subject of consciousness, including
quantum models. The incorrect perception that the quantum system has
only microscopic manifestations considerably confused this subject. As
we have seen in preceding sections, manifestation of ordered states is
of quantum origin. When
we recall that
almost all of the macroscopic ordered states are the result of quantum
field theory, it seems natural to assume that macroscopic ordered
states in biological systems are also created by a similar mechanism.
[8] We
argue herein that a scientific account of the secondary properties of
observation entails an overhaul of the traditional ontology of physics.
In this connection we spring Eugene Wigner,
whom Feynman called the most gifted physicist he ever met: "let us now
turn to the assumption opposite to the “first alternative” considered
so far: that the laws of physics will have to be modified drastically
if they are to account for the phenomena of life. Actually, I believe
that this second assumption is the correct one". Can
arguments be adduced to show the need for modification? There seem to
be two such arguments. The first is that, if one entity is influenced
by another entity, in all known cases the latter one is also influenced
by the former. The most striking and originally the least expected
example for this is the influence of light on matter, most obviously in
the form of light pressure. That matter influences light is an obvious
fact — if it were not so, we could not see objects. The
influence of light on matter is, however, a more subtle effect and is
virtually unobservable under the conditions which surround us ... Since
matter clearly influences the content of our consciousness, it is
natural to assume that the opposite influence also exists, thus
demanding the modification of the presently accepted laws of nature
which disregard this influence. [9] Why
should
we change the ontology of physics, though? Consider the following
passage from Michael Lockwood,
whose thought runs in tracks remarkably parallel to our own: Consciousness,
in other words, provides us with a kind of ‘window’ on to our brains,
making possible a transparent grasp of a tiny corner of a material
reality that is in general opaque to us, knowable only at one remove.
The qualities of which we are immediately aware, in consciousness,
precisely are some at least of the intrinsic qualities of the states
and processes that go to make up the material world — more
specifically, states and processes within our own brains. The
psychologist Pribram . . . has made an interesting attempt to revive an
idea originally put forward around the turn of the century by the
Gestalt psychologists: namely that it is certain fields, in the
physicist’s sense, within the cerebral hemispheres, that may be the
immediate objects of introspective awareness ... What it would amount
to, in terms of the present proposal, is that we have a ‘special’ or
‘privileged’ access, via some of our own brain activity, to the
intrinsic character of, say, electromagnetism. Put like that, the idea
sounds pretty fanciful. But make no mistake about it: whether about
electromagnetism or about other such phenomena, that is just what the
Russellian view ostensibly commits one to saying. There
are, however, two things that must now be emphasized. In the first
place, it is a clear implication of the Russellian view that the
material world, or more specifically, that part of it that lies within
the skull, cannot possess less diversity than is exhibited amongst the
phenomenal qualities that we encounter within consciousness. I am
inclined to doubt whether the stock of fundamental attributes
countenanced within contemporary physical science is, in principle,
adequate to the task of accounting for the qualitative diversity that
introspection reveals. The current trend, within physics, is towards
ever greater unification of the fundamental forces. [10] Lockwood
cited Pribram just now, apropos “certain fields, in the physicist’s
sense, within the cerebral hemispheres, that may be the immediate
objects of introspective awareness”. As Dyson has informed us, these
fields are quite fundamental to the physicist’s picture — a view
highlighted at the beginning of Hawking
and
Ellis’ work on The large scale structure of
space-time: "the
view of physics that is most generally accepted at the moment is that
one can divide the discussion of the universe into two parts. First, there is the question of the local laws
satisfied by
the various physical fields. These are usually expressed in
the
form of differential equations. [11] A
most explicit and philosophically sound expression of our working
scientific framework is found in a wonderfully erudite essay by Simon Saunders
on “The Algebraic Approach to Quantum Field Theory”: “Our basic
ontology is that all systems, macroscopic structures included, are
quantum fields . . .” [12] Finally,
and somewhat ironically, we note our agreement with Paul Churchland in
respect of: (a) the neural implementation of matrix operations on input
vectors; and (b) the observation, apropos the work of Pellionisz
and
Llinas (1979)*
that the cerebellum’s job is “the systematic transformation of vectors
in one neural hyperspace into vectors in another neural hyperspace”;
and the notion that (c) “the tensor calculus emerges as the natural
framework with which to address such matters” and (d) the
characterization of phenomenal properties as vectors. [13] If
Paul Churchland is correct about the neural implementation of
matrix-valued operators, then that is rather interesting, since that is
precisely the sort of mathematics we find at work at the quantum level
of neural function. Which would seem to make a kind of sense, if, as we
suggest, the form of neural networks follows the underlying function of
those quantum processes, which mediate neural activity. Given that that
the dendritic forms of neurons are aptly captured by the mathematics of
fractals, we might expect this kind of self-similarity across scales.
And so we might borrow from Paul to appease Patricia
[14] as to the relevance of QM to consciousness.† I
suspect we will continue to encounter a measure of not unintelligent
resistance from those who, perhaps not wishing to trouble themselves
with the admitted difficulties of quantum theory, may reply that, even
if we do have a remarkable correspondence between the mathematics of
neural networks and QFT (as would seem evident), we have no need to
pursue matters at the quantum level, because we can say all that needs
to be said at the neural level. We are not much moved by such replies,
which would seem to have more to do with intellectual inertia than with
logic or reason, let alone the spirit of scientific inquiry. For after
all, mind would seem to be connected to the brain somehow, and if the
matter of the brain resides at the quantum level, then the quantum
level would seem to be a logical place to look for mind. And then, who
can say before the fact what
discoveries might
await us in pursuing this line of research? Consider,
then, that QFT tells us that all is field phenomena. Dyson said so,
just now. Again, this would seem quite suggestive, when one reflects
upon the observation that the objects of our immediate awareness just
are perceptual fields. Then, too, we know our perceptual fields are
clearly related to those photonic fields which “excite” our perceptions
(as is most easily seen in the case of vision). We have two fields,
then, two sets of states, always varying together — mechanically,
predictably, quantifiably. Is this because the two are actually one?
Perhaps, but let us see whether we can make our proposal more
particular. 3)
Vectors Feynman
tells us that colors behave like vectors, which is interesting, because
photons happen to behave like vectors, too. Here is what Feynman wrote: The
second principle of color
mixing of lights is this: any color at all can be made from three
different colors, in our case, red, green, and blue lights. By suitably
mixing the three together we can make anything at all, as we
demonstrated... Further,
these laws are very interesting mathematically. For those who are
interested in the mathematics of the thing, it turns out as follows.
Suppose that we take our three colors, which were red, green, and blue,
but label them A, B, and C, and call them our primary colors. Then any
color could be made by certain amounts of these three: say an amount a
of color A, an amount b of color B, and an amount c of color C makes X:
X
= aA + bB + cC. Now
suppose another color Y is made from the same three colors:
Y = a'A + b'B + c'C. Then
it turns out that the mixture of the two lights (it is one of the
consequences of the laws that we have already mentioned) is obtained by
taking the sum of the components of X and Y: Z = X + Y = (a + a')A + (b + b')B + (c + c')C.
It
is just like the mathematics of the addition of vectors, where (a, b,
c) are the components of one vector, and (a', b', c') are those of
another vector, and the new light Z is then the "sum" of the vectors.
This subject has always appealed to physicists and mathematicians. In
fact, Schrödinger
wrote a
wonderful paper on color vision in which he developed this theory of
vector analysis as applied to the mixing of colors. [15] Now
let us
consider that mixing lights of different colors is accomplished by
superposing photonic state vectors. Here
is the luminous Weyl
to help us along with the physics: Monochromatic
light is completely determined as to its quality by the wave-length,
because its oscillation law with regard to time and its wave structure
have a definite simple mathematical form, which is given by the
function sine or cosine. Every physical effect of such light is
completely determined by the wave-length together with the intensity.
To monochromatic light corresponds in the acoustic domain the simple
tone. Out of different kinds of monochromatic light composite light may
be mixed, just as tones combine to a composite sound. This takes place
by superposing simple oscillations of different frequency with definite
intensities. The simple color qualities form a one-dimensional
manifold, since within it the single individual can be fixed by one
continuously variable measuring number, the wave-length. The composite
color qualities, however, form a manifold of infinitely many dimensions
from the physical point of view. [16] Weyl
writes “monochromatic
light is completely determined as to its
quality by the wave-length.”
But red and
green light, e.g., may be combined to give yellow. Weyl offers a way
out of this dilemma: This
discrepancy between the abundance of physical “color chords” and the
dearth of the visually perceived colors must be explained by the fact
that very many physically distinct colors release the same process in
the retina and consequently produce the same color sensation. By
parallel projection of space on to a plane, all space points lying on a
projecting ray are made to coincide in the same point on a plane;
similarly this process performs a kind of projection of the domain of
physical colors with its infinite number of dimensions on to the
two-dimensional domain of perceived colors whereby it causes many
physically distinct colors to coincide . . . It
seems useful to me to develop a little more precisely the “geometry”
valid in the two-dimensional manifold of perceived colors. [17] Since
we make so much of the vectorial aspects of color, let’s refresh our
memories as to the central role of vectors in the axioms of QM, with a
peek at a splendid modern text on the mathematics of classical and quantum physics: Axiom
I.
Any physical system is completely described by a normalized vector (the
state vector or wave
function) in
Hilbert space. All possible information about the system can be derived
from this state vector by rules (...) [18] As
to the superposition of state vectors, we read in another source: When
a state is formed by the superposition
of two other states, it will have properties that are in some vague way
intermediate between those of the original states and that approach
more or less closely to those of either of them according to the
greater or less 'weight' attached to this state in the superposition
process. The new state is completely defined by the two original states
when their relative weights in the superposition process are known,
together with a certain phase difference, the exact meaning of weights
and phases being provided in the general case by the mathematical
theory. [19] Is
the paragraph immediately above referring to the state vectors of QM?
Or to color vectors? The answer is the former, but notice that Dirac’s
remarks apply equally well to our common experience with mixing lights
of different colors. Years
ago, when I first broached this similarity of color vectors and QM
state vectors, a persistent comeback went something like, ‘So what?
Colors behave like vectors - lots of things behave like vectors, that
doesn’t prove anything.’ My considered reply was and is that, yes, many
things do behave like vectors. The point is, again, that color vectors
and photonic state vectors demonstrably co-vary in a manner altogether
predictable, reliable, and really quite quantifiable. A
second and related mathematical correspondence presents itself, one
which is both wonderfully simple and which goes to the heart of our use
of vectors in physical theory generally.
The
example of classical mechanics shows us that there are possible
representations of physical theories, which do not involve Hilbert
spaces. Of course, this doesn’t mean that classical mechanics could not
be reformulated in this way. In fact, our strategy for providing a
partial answer to the question, “Why Hilbert
spaces?” will be to show that the theory of vectors has very general
application. We will take as an example a particular physical situation
and model it mathematically. The situation will be paradigmatically of
the kind with which physical theory deals, but our description will be
general enough to leave open the question of what sorts of processes,
deterministic or indeterministic, are involved. Similarly, its
representation, in terms of vector space, will be general enough to be
employed for a variety of physical theories; the particular features of
quantum mechanics on the one hand, or classical mechanics on the other,
will then appear as additional constraints on these mathematical
structures, as proposed by Feynman: The key to the representation is the fact that Pythagoras’ theorem, or its analogue, holds in any vector space equipped with an inner product. Consider the space R^{3}. For any vector v in R^{3,} v = v_{x} + v_{y} + v_{z} _{ } Here v_{x,}
v_{y}, and
v_{z} are
the projections of v onto an orthogonal triple of rays spanning R^{3}—
or, as we can call them, the axes
of our coordinate
system.
Pythagoras’
theorem tells us
that |v_{x}|^{2}
+ |v_{y}|^{2}
+ |v_{z}|^{2}
= |v|^{2},
|v_{x}|^{2}
+ |v_{y}|^{2}
+ |v_{z}|^{2}
= 1
Let
us now assume that we wish to represent three mutually exclusive events
that together exhaust all possibilities, and that each event has a
certain probability. If we use the axes of R^{3} to
represent
the events x, y, and z, we can construct a normalized vector v to represent any probability assignment
to these events. We simply
take vectors v_{x},
v_{y}, and
v_{z} along these axes such that : |v_{x}|^{2}_{= p(x), }|v_{y}|^{2}_{ = p(y) and }|v_{z}|^{2}_{= p(z),} _{ } _{and then add them (vectorially) to yield v.} _{} Then considering the space R^{3}, for any vector v in R^{3:} v = v_{x} + v_{y} + v_{z” } that
takes
us full circle back to Feynman above, where any color vector v
can be produced by adding suitable amounts of red, green, and blue
vectors. So we have an obvious parallel between quantum states and
visual states. We might pursue this analogy and construct a normalized
color sphere with red, green, and blue serving duty as the projections
of v onto an “orthogonal triple of
rays” (RGB, e.g.)
spanning R^{3}, as in Hughes’ just quoted remarks. We
might also reflect on the fact of observation that no “point” in the
visual field can be more than one color simultaneously — though it must
have some color, as Wittgenstein remarks: "a speck in the visual field,
though it need not be red must have some color; it is, so to speak,
surrounded by color-space. Notes must have some pitch, objects of the
sense of touch some degree of hardness, and so on". [21] So:
a speck in the visual field must have (with probability 1!) some color.
It need not be red, but it must be some color, and cannot be more than
one color at the same time. So
we might
say each x, y, z, t of the visual field is "coordinated" with one color
and no other. Thus, the colors at each spacetime point are "mutually
exclusive and jointly exhaustive" — another essential parallel between
the states of the visual field and those of a quantum field. Now,
since any point in the visual field can be a different color from its
neighbor, it is rather as though each point in the visual field has an
“internal” color space attached to it — perhaps a color sphere such as
we have just now mentioned, with red, green and blue for its principle
axes. Curiously, this picture is strikingly similar to the mathematics
of gauge theory, with its algebra of “internal” spaces, an algebra that
determines the form of all particle interactions. Our
color sphere would also seem to resemble the compactified (very small)
orbifolds of string/M-theory, which are thought to sit over each point
in spacetime. So that’s kind of suggestive. But is the spacetime of
physics the same thing as the spacetime of perception? As Lindsay and
Margenau argue in their scholarly and companionable work on the foundations of physics, there
are important respects in which perceptual space and physical space
would seem not identical (though they also state, together with
Einstein, Mach, and others, that “public or physical space is an
abstraction by the mind of the aggregate of the various modes of sense
perception"). We
believe
that the answer to their reasonable concerns was foreshadowed just now
by Weyl, and is furthered in Poincaré,
where he writes: It is often
said that we
“project” into geometric space the objects of our external perception;
that we “localize” them. Has this a
meaning, and if so
what? Does it mean
that we represent to ourselves
external objects in geometrical
space? Our
representations are only the reproduction of our sensations; they can
therefore be ranged only in the same frame as these, that is to say, in
perceptual space. It
is as impossible for us to represent to ourselves external bodies in
geometric space, as it is for a painter to paint on a plane canvass
objects with their three dimensions. Perceptual
space is only an image of geometric space, an image altered in shape by
a sort of perspective [22] We
would
argue that our perceptual spaces are projections of a complete
spacetime, and in a wholly physical, quantum mechanical sense of
“projection”. Unfortunately, a discussion of the mathematics of the
“projection postulate” of QM would lead us too far afield here, though
we have touched upon it just now, in our discussion of Hilbert spaces
and the projection of v onto a
triplet of rays. We
would simply note that we are quite accustomed to using photons to
project images, in movies and television, e.g., and then of course the
visual world is projected onto our retinas by photons, which retinal
image is then projected to other areas of our brains by processes
“electromagnetic in origin”. And then, of course, in projecting an
image on a screen, we routinely project the colors of that image. Let’s
look
in on Weyl again and see what he has to say about colors and projective
geometry: Mathematics
has introduced the name 'isomorphic representation' for the relation,
which according to Helmholtz exists between objects and their signs. I
should like to carry out the precise explanation of this notion between
the points of the projective plane and the color qualities...On the one
side, we have a manifold S_{1}
of objects — the points of a convex section of the projective plane —
which are bound up with one another by certain fundamental relations R,
R’, ...; here, besides the continuous connection of the points, it is
only the one fundamental relation: 'The point C
lies
on the segment AB'.
In projective geometry no notions occur except such as are defined on
this basis. On the other side, there is given a second system S_{2} of objects
— the manifold of
colors — within which certain relations R, R’,
. . . prevail which shall be associated with those of the first domain
by equal names, although of course they have entirely different
intuitive content. Besides the continuous connection, it is here the
fundamental relation: 'C arises by a
mixture from A and B';
let us therefore express it
somewhat strangely by the same words we used in projective geometry:
'The color C lies on the segment
joining the colors A and B'. If now the elements of the
second system S_{2} are made to
correspond to
the elements of the first system S_{1 }in such a
way, that to
elements in S_{1 }for which
the relation R, or R’,
or . . . holds, there always correspond elements in S_{2 }for which
the homonymous
relation is satisfied, then the two domains of objects are
isomorphically represented on one another. In
this sense the projective plane and the color continuum are isomorphic
with one another. Every theorem which is correct in the one system S_{1 }is transferred
unchanged to the other S_{2} (Our
emphasis) [23]. Weyl
says that the “projective plane and the color continuum are isomorphic
with one another.” It may be worthwhile to reflect on the central role
of such dualities in string/M-theory and to consider that the
compactified orbifolds of that theory of are known to be projective
spaces. Then, too, the Riemannian geometry of relativity is known to be
a special case of projective geometry, as Kline relates: It
became possible to affirm that projective geometry is indeed logically
prior to Euclidean geometry and that the latter can be built up as a
special case. Both Klein and Arthur Cayley showed that the basic
non-Euclidean geometries developed by Lobachevsky and Bolyai and the
elliptic non-Euclidean geometry created by Riemann can also be derived
as special cases of projective geometry. No wonder that Cayley
exclaimed, "Projective geometry is all geometry." [24] Are
perceptual fields quantum fields? Well, maybe, but there are other
reasons to call upon quantum theory in discussions of consciousness,
for as Lockwood related just now, “it is a clear implication of the
Russellian view that the material world, or more specifically, that
part of it that lies within the skull, cannot possess less diversity
than is exhibited amongst the phenomenal qualities that we encounter
within consciousness”. Later, he writes: “If mental states are brain
states, then introspection is already, it seems to me, telling us that
there is more to the matter of the brain than there is currently room
for in the physicist’s philosophy.” Just
so. And here is the crux of our problem: remember that we are routinely
told that the wavefunction provides the most complete description of
the system that is, in principle, possible. As is well known, Einstein
found all this quite implausible: “one
arrives at very implausible theoretical conceptions, if one attempts to
maintain the thesis that the statistical quantum theory is in principle
capable of producing a complete description of an individual physical
system.” [25] But
so did
the author of the Schrödinger wave function: If
you ask a physicist what is his idea of yellow light, he will tell you
that it is transversal electromagnetic waves of wavelength in the
neighborhood of 590 millimicrons. If you ask him: But where does yellow
comes in? he will say: In my picture not at all, but these kinds of
vibrations, when they hit the retina of a healthy eye, give the person
whose eye it is the sensation of yellow. [26] 4)
A little knowledge There
are those who may object at this point, saying something like ‘but
everybody knows that color is just the frequency (or wavelength) of
light.’ To which we reply that Schrödinger seems to have known a thing
or two about physics and was, moreover, the leading authority on color
science in his day, so we might ask how this fact escaped him? Then
again, if color has been identified as somehow being equal to (say)
wavelength, how was this identity established? And by whom? And when?
And why is color sometimes identified with the frequency of light,
whereas other authorities casually identify color with wavelength or
energy? And why are such well-established facts nowhere supported in
the literature of physics? And how is it possible to write entire texts
on optics and never once employ the word, “color”? More importantly for
science, since a wavelength is a length, and therefore a scalar
quantity, why does color behave like a vector quantity? The
answer
can be found in the classic works of Burtt, Duhem, and Lovejoy, but
Weyl puts the matter succinctly: The
idea of the merely subjective, immanent nature of sense qualities, as
we have seen, always occurred in history woven together with the
scientific doctrine about the real generation of visual and other sense
perceptions…Locke’s standpoint in distinguishing primary and secondary
qualities corresponds to the physics of Galileo, Newton, and Huyghens;
for here all occurrences in the world are constructed as intuitively
conceived motions of particles in intuitive space. Hence an absolute
Euclidean space is needed as a standing medium into which the orbits of
motion are traced. One can hardly go amiss by maintaining that the
philosophical doctrine was abstracted from or developed in close
connection with the rise of this physics. [27] Let
us pause to sweep away a few centuries of dust. Rather than indulge the
herd, we prefer to follow the lead of Maxwell, Schrödinger, Russell,
Weyl, Einstein et al., all of whom devoted serious thought to these
matters, and generally dismiss what “everyone knows” as a kind of
shorthand for “so much sloppy thinking”. We prefer to point out the
historical facts, and foremost among these must be the fact that the
“hard problem” flows directly from the old division of the elements of
perception into two camps, viz., the primary and the secondary.
According to this doctrine, bequeathed to us by the fathers of science,
such “primary” qualities as extension in space and duration in time are
thought to inhere in material objects themselves, to be “physical
quantities” as we now say. Whereas such “secondary” qualities as color
and sound have long been thought to exist only “in the mind”, perhaps
as a result of the movements of the brain's material constituents —
even though those constituents have, by
definition, no
such secondary properties. This
old division, of primary and secondary, has its roots in Euclid and the
Greek atomists, but it was Kepler, Galileo, Newton and Boyle, together
with Descartes, Locke, Hobbes, and that crew who enshrined this duality
at the foundations of physical theory. So the doctrine has a fine
pedigree, and persists in our own time, in a largely unexamined light,
as part of what “everyone knows” - though as scientists, our first
question must be: is the division of primary and secondary true to
nature? A modern statement of the primary/secondary doctrine, and a
telling response to it, are found in Einstein’s essay “On Russell’s
Theory of Knowledge”. Einstein quotes Russell: We
all start from 'naive realism,' i.e., the doctrine that things are what
they seem. We think that grass is green, that stones are hard, and that
snow is cold. But physics assures us that the greenness of grass, the
hardness of stones, and the coldness of snow, are not the greenness,
hardness, and coldness that we know in our own experience, but
something very different. The observer, when he seems to himself to be
observing a stone, is really, if physics is to be believed, observing
the effects of the stone upon himself. Thus science seems to be at war
with itself: when it means to be most objective, it finds itself
plunged into subjectivity against its will. Apart
from their masterful formulation these lines say something which had
never previously occurred to me. For, superficially considered, the
mode of thought of Berkeley and Hume seems to stand in contrast to the
mode of thought in the natural sciences. However, Russell's just cited
remark uncovers a connection: If Berkeley relies upon the fact that we
do not directly grasp the "things" of the external world through our
senses, but that only events causally connected with the presence of
"things" reach our sense-organs, then this is a consideration which
gets its persuasive character from our confidence in the physical mode
of thought. [28] What
aspect(s) of the “physical mode of thought” might be at fault? We would
urge that the ontology of QM is prima facie
incomplete, and in the sense of EPR, for not every “element of reality”
is represented within the theory — namely, greenness, hardness, and
coldness, together with the other secondary properties. Such
statements as the foregoing are apt to result in hands waving, tempers
flaring, and pedants thundering in an impressive display of cognitive
dissonance. All very entertaining, but typically not terribly
enlightening. Alas, the issue will not go away, for the textbooks tell
us that all we can know about a state is contained “in principle” in
Schrödinger’s equation, and yet Schrödinger himself says otherwise.
Then, too, many of our most stellar scientists past and present have
pondered these issues, and with divergent results. The moral would seem
to be that while we are certainly in good respectable scientific
company here, these foundational issues are by no means settled, and so
a measure of controversy is to be expected, if not deliberately incited. The
fact that so many intelligent, scientifically minded people will try to
dodge the mind/body issues altogether, however, might suggest that we
are, perhaps, not all that confident in the “physical mode of thought”
when it comes to greenness and coldness and hardness and so forth. Then
again, there would seem to be those who are altogether too confident in
that mode of thought — or, at any rate, its standard ontology. How else
to explain those learned persons who reason that, if colors have no
place in science, then colors must be nonexistent or illusionary? At a
glance, and put like that, such a stance must strike us as absurd. But
celebrated scholarly works have been written in recent years which make
just such arguments, albeit in more sophisticated terms, and though the
absurdity of the conclusion remains the same. But we needn’t be too
harsh on our esteemed colleagues. For as an eminent contemporary
authority argues, this old doctrine of Newton and company, this divorce
between the primary and secondary qualities, made practical good sense
at the time, when our fledgling science was just getting on its feet,
though it obviously raises a number of serious issues once we attempt
to discover a science of consciousness: The
world as described by natural science has no obvious place for colors,
tastes, or smells. Problems with sensory qualities have been
philosophically and scientifically troublesome since ancient times, and
in modern form at least since Galileo in 1623 identified some sensory
qualities as characterizing nothing real in the objects themselves... The
qualities of size, figure (or shape), number, and motion are for
Galileo the only real properties of objects. All other qualities
revealed in sense perception — colors, tastes, odors, sounds, and so on
— exist only in the sensitive body, and do not qualify anything in the
objects themselves. They are the effects of the primary qualities of
things on the senses. Without the living animal sensing such things,
these 'secondary' qualities (to use the term introduced by Locke) would
not exist. Much
of modern philosophy has devolved from this fateful distinction. While
it was undoubtedly helpful to the physical sciences to make the mind
into a sort of dustbin into which one could sweep the troublesome
sensory qualities, this stratagem created difficulties for later
attempts to arrive at some scientific understanding of the mind. In
particular, the strategy cannot be reapplied when one goes on to
explain sensation and perception. If physics cannot explain secondary
qualities, then it seems that any science that can explain secondary
qualities must appeal to explanatory principles distinct from those of
physics. Thus are born various dualisms. [29] Can
physics explain secondary qualities? It seems it must attempt to do so,
for our scientific picture of the world rests on sense perception, and
agreement with observation must be the ultimate basis of the empirical
content of science. And sense perception regularly, mechanically
discloses to us these secondary properties of color and sound and so
forth, and in predictable spacetime association with physical,
(quantum!) mechanical stimuli. And yet, as Schrödinger tells us, these
properties have no agreed-upon place in the formalisms of traditional
physics. So
it is, then, that we find ourselves still in Leibniz' mill, unable to
explain such a simple thing as a perception of yellow in respect of the
brain's machinery — essentially the same problem we find in Penrose'
challenge to strong AI: Where in all the software and circuitry is the
mind? And again in Chalmers: If the brain is a physical thing, composed
of particles which, by definition, contain no yellow, then what is the
yellow doing there? And
this is the crux of the mind/body duality, as Hume understood in his
day, when the whole scam was going down, though we have yet to learn
the lesson today: The
fundamental principle of that philosophy is the opinion concerning
colors, sounds, tastes, smells, heat and cold; which it asserts to be
nothing but impressions in the mind, deriv’d from the operation of
external objects, and without any resemblance to the qualities of the
objects. This
principle being once admitted, all other doctrines of that philosophy
seem to follow by an easy consequence. For upon the removal of sounds,
colors, heat, cold, and other sensible qualities, from the rank of
continu’d independent existences, we are reduced merely to what are
called primary qualities, as the only real ones, of which we have any
adequate notion. . . . Thus
there is a direct and total opposition betwixt our reason and senses...
When we reason from cause and effect, we conclude, that neither color,
sound, taste, nor smell have a continued and independent existence.
When we exclude these sensible qualities there remains nothing in the
universe, which has such an existence. [30] Back
to the present, we have said that the answer to the duality of mind and
body is contained in its essentials in Chalmers and Feigl. Now we see
that we must deal with the duality of primary and secondary, for in the
case of the visual field and its concomitant photon field, we want to
argue that the two information states, of color and photon, are
something like distinct but complementary aspects of a single, more
complete state. How to resolve the duality of primary and secondary? 5) More light Let us
attend Mach, where, in
his Analysis of Sensations.
he writes, in open contradiction to Galileo and his gang: “A color is a
physical object as soon as we consider its dependence, for instance,
upon its luminous source, upon temperatures, upon spaces, and so
forth.” [31] We
seem to have Mach’s OK so far as treating color as a physical, and not
a mental, thing. His reasoning seems quite sound to us, though Mach was
criticized in his time for confusing ‘things’ with our ‘perceptions of
things’. In
what follows we cheerfully ignore Mach’s critics and, for the sake of
argument, generally assume the truth of his just quoted remarks, and
see where that might take us. In fact, Mach’s argument regarding the
physical nature of color (and, by extension, sound, etc.,) contains the
only truly fundamental change in physical theory required herein. Now,
there is no escaping the fact that it is a fundamental change, for as
Hume noted just now, our entire scientific world view flows from this
one crucial divorce between primary and secondary. In repairing this
rift, we hope to achieve a unity, harmony, and identity between mind
and brain, but we cannot even mention all the many possible objections
that might be raised to our course in this brief space. Rather, we take
the stance that the proof is in the pudding, and argue that the
inclusion of the secondary properties among the elements of physics
makes for a more consistent, coherent, and complete account of nature.
But since the physical nature of color is such a very important point,
perhaps a few words would be in order by way of justifying our position. Why
should color not be physical? Well, the color of a thing is supposedly
subjective, whereas the length of a thing, e.g., is surely an objective
aspect of that thing. Yet relativity tells us that both the length and
the color of a thing are dependent upon the observer’s state of motion
— and in a perfectly objective way. Can relativity help us out here?
Indeed it can, as the redoubtable Weyl relates: The
immediately experienced is subjective but absolute; no matter how
cloudy it may be, in this cloudiness it is something given thus and not
otherwise. To the contrary, the objective world which we continually
take into account in our practical life and which science tries to
crystallize into clarity is necessarily relative; to be represented by
some definite thing (numbers or other symbols) only after a system of
coordinates has been arbitrarily introduced into the world. We said at
an earlier place, that every difference in experience must be founded
on a difference of the objective conditions; we can now add: in such a
difference of the objective conditions as is invariant with regard to
coordinate transformations, a difference that cannot be made to vanish
by a mere change of the coordinate system used ... [32] Let’s
pay close attention to what Weyl said just now about “every difference
in experience must be founded on a difference of the objective
conditions” and “in such a
difference of the
objective conditions as is invariant with regard to coordinate
transformations” Why is this important? Well, relativity flows from the
invariance of the laws of nature with regard to coordinate
transformations, and we now know that such invariances or symmetries
inform the very foundations of all physical theory. So we might do well
to attend to the fact that the color of a thing, e.g., remains the same
under translations, rotations, and reflections in spacetime — other
things being equal. Or as Helmholtz put it, with brilliant clarity:
“Similar light produces under like conditions a like sensation of
color.” 6) Symmetry Now,
in quoting Helmholtz on this point in the past, I have had not
uneducated persons object that, if you change the lighting, or the
room, the color changes. On hearing such replies, I am now inclined to
think that the other party (a) is not heeding the clause concerning
“under like conditions”; and (b) is not yet fully cognizant of the
import of her words; and (c) constitutes part of a scientific culture
which strains under a large burden of ignorance concerning the
secondary properties. So what? Well, it could scarcely be more
important. For yes, colors do change when the light changes — and
colors always change in the same way, here or there, now or then,
always and everywhere, world without end. It is very like a law of
nature. It is quite a lot like an invariance or symmetry of nature,
such as those now known to determine the evolution of the state vector
in QM. Furthermore,
and now this is the point, this is the punch line, the symmetries
determine the action. This action, this form of the dynamics, is the
only one consistent with these symmetries ... This, I think, is the
first time that this has happened in a dynamical theory: that the
symmetries of the theory have completely determined the structure of
the dynamics, i.e., have completely determined the quantity that
produces the rate of change of the state vector with time. This
sounds interesting, but a natural question arises: What are these
symmetries Weinberg speaks of, precisely? Earlier on, he writes: Increasingly,
many of us have come to think that the missing element that has to be
added to quantum mechanics is a principle, or several principles, of
symmetry. A symmetry is a statement that there are various ways that
you can change the way you look at nature, which actually change the
direction the state vector is pointing, but which do not change the
rules that govern how the state vector rotates with time. The set of
all these changes in point of view is called the symmetry group of
nature. It is increasingly clear that the symmetry group of nature is
the deepest thing that we understand about nature today. [33] This
sounds rather abstract, perhaps, but the symmetries Weinberg speaks of
are directly akin to the usual sorts of symmetries familiar to us from
daily observation, as well as from the most basic tenets of relativity: The
meaning of symmetry of a physical system is frequently influenced, if
not shaped, by the guidelines of our investigation. It
is obvious that the symmetry of a physical system is closely related to
the transformations of the parameters describing the system. Notice,
however, that not every transformation of parameters is linked to a
symmetry of the system; such symmetries have to satisfy certain
conditions. The necessary condition is that the physical system remains
the same object of our perception before as well as after the
transformation . . . We
say that a 3-dimensional sphere has a rotational symmetry because the
picture of it does not change while we rotate it through an angle
around an arbitrary axis through the center of this sphere. . . Let us
take as an example the relativistic field theory. This theory as a
whole is symmetric with respect to the Lorentz transformations. This
means that independently of the choice of frame of reference, the same
field theory is the object of our investigation; changing from one
frame to another the fields transform covariantly according to the rule
imposed by the principle of relativity. [34] Let’s
carefully consider the following bit, in relation to perception: “The
necessary condition is that the physical system remains the same object of our perception before as well
as after the
transformation.” (Our emphasis) We
are moved to reflect that, other things being equal, the color of a
laser (say) does not change as a result of moving the laser about in
spacetime. Similarly, other things being equal, a tuning fork will
sound the same, here or there, now or then, always and everywhere. So
we might ask: Do the symmetries of color and sound and so forth also
contribute to the “structure of the dynamics”? Perhaps, but we want to
go a bit further and tighten up our thesis as to how such simple
invariant properties as “yellow” might be incorporated into the body of
quantum theory. Before doing so, we really must have a glance at
Noether’s theorem: “If the Schrödinger equation has a symmetry with
respect to some unitary group, then the observable corresponding to the
self-adjoint generator of the group is a constant of motion.” On this
slightly more technical note, we point out that the color vector
associated with the Schrödinger state vector of a photon would seem to
be symmetric — a “constant of motion” — with respect to the group of
translations, rotations, and reflections. 7)
Simplify, simplify Let
us attend to the simplicity of colors. For colors are so simple, we
might think of them as elemental, and so perhaps count them among the
proper elements of an EPR-complete quantum theory. What does this mean?
Let’s remind ourselves of what Einstein & Co., said in their
seminal work on the (in)completeness of QM: In
attempting to judge the success of a physical theory, we may ask
ourselves two questions: (1) “Is the theory correct?” and (2) “Is the
description given by the theory complete?” It is only in the case in
which positive answers may be given to both of these questions, that
the concepts of the theory may be said to be satisfactory. The
correctness of the theory is judged by the degree of agreement between
the conclusions of the theory and human experience... Whatever
the meaning assigned to the term complete, the following requirement
for a complete theory seems to be a necessary one: every element of the
physical reality must have a counterpart in the physical theory. [35] We
see that our investigation naturally leads us to the question: Are the
secondary properties among the “hidden variables” of QM? Well,
“everybody knows” hidden variables are generally out of favor at the
moment. But a glance at the history of modern physics will reveal that
hidden variables theories have never been in vogue. Holland gives an
account of the sober scientific reception with which discussion of
hidden variables has historically been greeted. The
reaction to Bohm’s work by the Copenhagen establishment was generally
unfavourable, unrestrained and at times vitriolic (e.g. Rosenfeld
(1958)). We
first consider the response of Heisenberg (1955; 1962, Chap. 8) to
Bohm’s contribution, and see in outline how the points he raises may be
answered (see also Bohm (1962) for a reply). Heisenberg first questions
what it means to say that a wave propagating in configuration space is
‘real’. His objection to this notion is based on his assertion that
only ‘things’ in three-dimensional space are ‘real’. He offers no
logical or scientific argument to show that examining the possibility
of multidimensional spaces is a fruitless enterprise. It is useful to
recall here the Kaluza-Klein program in general relativity where
physicists contemplate spacetimes of dimension greater than four as a
valuable aid to comprehending and unifying the basic physical
interactions. The
‘hidden parameters’, i.e., the particle orbits, are denounced by
Heisenberg as a ‘superfluous ‘ideological superstructure’” having
little to do with immediate physical reality because the causal
formulation generates the same empirical results as the Copenhagen
view. But in Bohm’s theory it is precisely
the positions
of particles that are recorded in experiments; they are the immediately
sensed ‘reality’. [36] Well,
so
what of these “hidden” variables or parameters? What are they,
precisely? A
“hidden-variable” theory, as the name implies, postulates that
alongside (or, more graphically, beneath) the measurable quantities
dealt with by the theory (position, momentum, spin, and so on) there
are further quantities inaccessible to measurement[1],
whose values determine the values yielded by individual measurements of
the observables. The quantum mechanical statistics are to be obtained
by “averaging” over the values of the hidden variables. The
inaccessibility of these variables may be a contingent and temporary
matter, to be remedied as we develop new experimental procedures, or
these quantities may be in principle inaccessible (see Jammer, 1974, p.
267). The
suggestion that there may be such “hidden variables” is as old as the
probabilistic interpretation of the state vector. It was made by Born
(1926b, p. 825) a few months after he first proposed that
interpretation: “Anyone dissatisfied with
these ideas may
feel free to assume that there are additional parameters not yet
introduced into the theory which determine the individual event" [37].
But almost as old is the denial that such hidden variables can exist. 8)
Manifest variables It
is well worth noting that Born’s just cited remark, that we “may feel
free to assume that there are additional parameters not yet introduced
into the theory which determine the individual event” is rarely to be
found in the standard literature on the subject. Having attended for
many years the ongoing discussions among those more serious thinkers
who are little concerned with intellectual fashion, we are inclined to
agree with Gell-Mann, who writes of Bohr and Heisenberg having
“brainwashed” a generation of physicists. While we have no interest in
generating a diatribe, it is at least curious to consider mainstream
physicists embracing many unobserved worlds in order to skirt the
observer problem, even if this move entails throwing Occam out the
window. Whereas we have daily experience of (secondary) variables
which, though “hidden” by dogma, are nonetheless in observed
association with those quantum mechanical things known as photons.
Happily, according to Wheeler and Tegmark [38] the reign of Copenhagen
appears to have come to an end, and so perhaps it is not too much to
hope for a more reasoned and informed debate of foundational issues - something beyond the “shut up and
calculate”
approach. Now,
there are also not a few today who, upon reading these words, will
object and say: "yes, but our perception of yellow depends on the
neurological state of the observer, therefore color is 'subjective',
and therefore a subject for neurology, and not for physics". Well, this
is only Galileo warmed over, but all right, we can reply that our
perceptions of space and time are also dependent on the observer’s
neurological state. Are space and time therefore subjective? Well, no
... for scientific purposes we agree on units of space and time — the
rigid rods and standard clocks of relativity, e.g.. But we also employ
such standards with respect to colors and sounds. Our color televisions
and stereos physically incorporate such standards, and in fact we know
quite a lot about how to add waveforms in order to produce this color
or that sound. With respect to a future geometry of color and sound, we
are arguably today in something like the situation which obtained with
Euclidean geometry before Euclid came along — we have quite a lot of
practical knowledge available, but no unifying theory. As
for color perception being a problem for neurology, one could also
point out that neurons are, presumably, physical things, and therefore
subject matter for physics. But I have a sneaking suspicion that those
who argue that color is not a problem for physics are not really
troubled by issues of reason, logic, or history so much as by the
(rather daunting) prospect of overhauling the standard ontology of
physics — what naturally strikes many as a radical move, somehow going
against the grain of what they take physics to be about.
I
believe that many physicists and their admirers have a problem in their
gut with a physics that is full of greenness, hardness, and coldness —
and they have centuries of accumulated scientific success to back them
up, together with the weight of illustrious authority, which has stated
from the outset that colors and sounds do not belong to the world of
physics. Now, one could counter such objections by simply noting that
colors and sounds are observed properties of the world, and it is the
business of physics to account for such properties and their manifest
regularities, and never mind the appeals to authority, or trying to
sweep the problem under the rug of mind, metaphysics, or neuroscience,
because we’ve tried all that and it simply doesn’t work. And these
would be reasonable replies, up to a point. Then
again, to prevail against authority we can do more than patiently argue
the facts. We can bring in other, equally weighty authorities to
reassure us that we are on safe, intellectually respectable premises.
We can call upon the likes of Leibniz, Hume, Young, Helmholtz, Mach,
Maxwell, Weyl, Schrödinger and Einstein, all of whom devoted serious
thought to these matters — as did Newton himself, the father of color
science (as well as a physicist of no small stature), who wrote that
“the science of colors becomes a speculation as truly mathematical as
any other part of physics.” [39] A
more cogent concern from standard physics might be: QFT is our most
precise science, and its many successes are legendary. How would
enlarging upon the ontology of QFT help in any way? The surprising but
natural reply is that the inclusion of the secondary properties of
color and sound and so forth would make for a more complete
physical theory. Where, again, “complete” is used in the sense of EPR,
where every “element of reality” is represented in the theory. Again,
this move amounts to an identification of the secondary properties of
observation with the hypothetical and much discussed “hidden variables”
of QM. (Which identification would amount to an extraordinary irony,
since colors and sounds are not “hidden” from us, except in plain sight
and within hearing.) But
again, as everyone knows, local hidden variables theories are thought
to have been dealt a death blow by the ingenious experiments of Aspect
et al., and so this loose identification of secondary properties with
hidden variables would seem not very promising at the outset, and,
worse, might serve only to
complicate matters
where no such help is needed, thank you very much. Our
reply is manifold: (1) we are in sympathy with Bell, who wrote that
what “no HV theories” demonstrate is a lack of imagination; (2) we seem
to be stuck with nonlocality anyway, so the door would appear to be
open just there — perhaps nonlocal communication occurs via the space
of secondary properties? for (3) colors are nonlocal and atemporal in
the sense that two photonic vectors of identical energy will exhibit
identical color vectors for the standard observer — thus, even though
the photons be lightyears apart, they would nonetheless appear to
occupy the same “point” in color space; and (4) proving #3 quite
suggestively requires the same precise mathematics employed by
relativity and gauge theory generally, i.e., parallel transport in a
curved spacetime; and then (5) the compactified dimensions of
string/M-theory are well accepted, and yet these additional spatial
variables are, at present, “hidden”; whereas (6) the secondary
properties are manifestly spatio-temporal in the sense that they
predictably intersect spacetime — much like the compactified dimensions
of string/M-theory are supposed to do; and (7) in an empirical science,
in which sensory experience must be the final court of appeal, the
existence of observed entities must carry more force than rationalistic
arguments (such as Von Neumann’s) which purport to deny the possibility
of such entities; whereas (8) the theory of real numbers is very
beautiful and powerful, but the theory’s beauty and power are only
enhanced by the addition of the complex numbers; and, (9) a coherent
scientific account of the secondary properties and of consciousness
generally would constitute nothing less than a major advance in our
physical understanding of the natural world in
toto; and,
finally, (10)
physicists already make routine use of colors in describing such
physical things as blue dwarfs, red shifts, gold and silver, and so on. And
indeed Mendeleev, in constructing the periodic table, was guided by the
secondary properties of the elements, and so, in a sense, the present
proposal only amounts to giving formal recognition to variables already
in daily scientific use. Polemics
aside, it would seem as though we have on one hand the secondary
properties, looking for a home in the formalisms of physics, whereas on
the other hand we have these empty spaces in various contemporary
physical theories. So we want
to see whether we
can "coordinate" these empty slots with the homeless secondary
properties - in the case of color and vision, we want to see whether we
can match the states of the photon field with the states of the
associated visual field. 9)
Manifolds Let’s
pick up the mathematical line again, and remind ourselves of Riemann,
who, at the beginning of his famous habilitation lecture, points out
that colors and space are both manifolds: So
few and far between are the occasions for forming notions whose
specialization make up a continuous manifoldness, that the only simple
notions whose specialisation form a multiply extended manifoldness are
the positions of perceived objects and colors. More frequent occasions
for the creation and development of these notions occur first in the
higher mathematics. Definite
portions of a manifoldness, distinguished by a mark or a boundary, are
called Quanta ... [40] The
upshot
would seem to be as follows: We have two states consisting of two kinds
of vectors occupying two manifolds. And
the two
always and everywhere co-vary. Which leads us back to Weyl: In
the realm of physics it is perhaps only the theory of relativity which
has made it quite clear that the two essences, space and time, entering
into our intuition, have no place in the world constructed by
mathematical physics. Colors are thus “really” not even
aether-vibrations, but merely a series of values of mathematical
functions in which occur four independent parameters corresponding to
the three dimensions of space, and the one of time. [41] Notice
that Weyl has no trouble coordinating colors with spacetime. Notice,
too, though, the tendency to diminish the role or nature of color:
“Colors are “really” not even aether-vibrations” but “merely” values of
functions. Our essential departure with much traditional thinking along
these lines consists in regarding colors and so forth as elemental,
though of course in saying so we are well within the tradition found in
Maxwell, Wittgenstein, Weyl, and Russell & Whitehead. So,
what of these theories with their additional dimensions? What can we
say for sure about them? First, a bit of background from one of the
master architects of string theory: I’m
sympathetic to the view that these theories are at present very remote
from being able to explain directly what is measured experimentally in
accelerator laboratories. Given the fact that they are so very
different from previous kinds of theories, then they ought to predict
some entirely new sort of phenomenon that we haven’t even thought of
measuring. It was only after
Einstein had formulated
general relativity that he understood which phenomena that could be
measured, would test the theory. The precession of the perihelion of
the planet Mercury was already known, but it wasn’t until Einstein came
up with general relativity that it was realized that this peculiar
anomaly was of fundamental importance. So what we need in superstring
theory is the analogue of the planet Mercury. Some
distinctive piece of experimental evidence that might already be known
but hasn’t struck anyone as being important because no one realizes
that it’s of relevance to testing a fundamental theory. (Our
emphasis) [42] On
the present view, should the secondary properties turn out to reside in
the additional spatial dimensions of something like M-theory, then that
would seem to meet the foremost objection to string/M-theory, that it
does not meet up with observation. Green
also writes, “Well, obviously the extra dimensions have to be different
somehow because otherwise we would notice them.” This is curiously
parallel to a remark from David Bohm: “Now it may be asked why these
hidden variables should have so long remained undetected.” John
Schwarz, another among the principal architects of string/M-theory,
provides us with a measure of context and motivation regarding the
additional dimensions: If
we knew what that six-dimensional space looked like we would be in a
great position for calculating all sorts of things that we want to
know. This may sound surprising. After all, as I have already said,
this space is completely invisible because it’s too tiny to observe
directly. As it turns out the details of its geometry and topology
actually play a crucial role in determining the properties of
observable particles at observable energies. [43] On
the present view, the just cited remarks from Bohm, Green, and Schwarz
must strike us as quite ironic, given that we do notice or detect
colors and sounds all the time, as a consequence of the only directly
observable particles we know — i.e., photons. So why have the secondary
properties not been put forward heretofore to occupy these “hidden”
variables and extra dimensions? Part of the answer must lie in the fact
that colors and sounds have historically been excluded from the
physical world, even though they demonstrably co-vary with other
physical parameters. Another part of the answer is contained in an
observation from Wittgenstein, where he writes that “the things that
are most important for us are hidden from us by their simplicity and
familiarity.” And then, of course, the dimensions of color and sound
and so forth are different from the dimensions of traditional
spacetime; they are more like the “internal” dimensions of gauge theory
or the compactified (very small) dimensions of string/M-theory — and
like these more traditional physical dimensions, the dimensions of
color and sound are tangent to the points of spacetime, suggesting that
colors and sounds might be amenable to the mathematics of fiber bundles. Another
important class of field theories, having a familial relation to
M-theory, is the Kaluza-Klein theories: There
are other kinds of unitary field theories, including some that today
claim a great deal of interest. These utilize, in some way or other, an
increase in the number of dimensions of space-time. One famous example
is Kaluza's proposal. He increased the number of dimensions to five,
without changing the Riemannian character of the model. He was thus
able to increase the number of components of the metric so as to
accommodate the electromagnetic field as well. He set one extra
component equal to a constant, because he had no use for it. To account
for the observed four-dimensionality of space-time, he assumed that no
field depended on the fifth coordinate. [44] Note:
“To account for the observed four-dimensionality of space-time, he
assumed that no field depended on the fifth coordinate.” But the
spacetime of our perceptions is not four-dimensional, for to each point
in the visual field, e.g., must correspond a color, which requires an
additional parameter or dimension. The assumption that the additional
spatial dimensions must be exceedingly small because we do not "see"
them dates back to the seminal work of Kaluza: Although
all our previous physical experience hardly provides any suggestion of
such an extra world-parameter, we are certainly free to consider our
space-time to be a four-dimensional part of an R5; one then has to take
into account the fact that we are only aware of the space-time
variation of quantities, by making their derivatives with respect to
the new parameter vanish or by considering them to be small as they are
of higher order ... [45] Cao
brings out the inter-relatedness of the extra dimensions of gauge and
M-theory, together with the mathematics of fiber bundles: Just
as in the original spinning string model, [the] superstring also
requires ten-dimensional space-time. So, to be of relevance to physics,
the extra six dimensions must compactify and be very small . . . From
the
above brief review, we find there are three versions of geometrization
of non-gravitational gauge interactions: 1.
Fibre-bundle version, in which the gauge interactions are correlated
with the geometrical structures of internal space. Since it is possible
to get a non-trivial fusion of space-time with internal space, the
gauge interactions also have some indirect relation with space-time
geometry. But the essence of the internal space is still a vexing
problem: Is it a physical reality as real as space-time, or just a
mathematical structure? 2.
Kaluza-Klein version, in which extra space dimensions which compactify
in low-energy experiments are introduced and the gauge symmetries by
which the forms of gauge interactions are fixed are just the
manifestation of the geometrical symmetries of the compactified space.
Here the mediator between the gauge interactions and the space-time
geometry is no longer the vexing internal space but the real though
compactified extra space dimensions. The assumption of the reality of
the compactified space is substantial and is in principle testable,
although its ad-hoc-ness makes it difficult to differentiate it from
the internal space in the fibre-bundle version. 3.
Superstring version, in which the introduction of extra compactified
space dimensions is due to different considerations from just
reproducing the gauge symmetry. Therefore, the properties and
structures of the compactified dimensions are totally different from
those in the Kaluza-Klein version. For example there is no symmetry in
the compact dimensions from which the gauge symmetries emerge; the
gauge interactions are correlated with the geometrical structure of
ten-dimensional space-time as a whole but not just with the extra
dimensions. [46] OK,
but are we really justified in thinking of secondary properties as
dimensions? Let us attend Minkowski in his famous essay on the unity of
space and time: We
will try to visualize the state of things by the graphic method. Let x,
y, z be rectangular co-ordinates for space, and let t denote time. The
objects of our perception invariably include places and times in
combination. Nobody has ever noticed a place except at a time, or a
time except at a place. . . . The multiplicity of all thinkable x, y,
z, t systems of values we will christen the world. [47] We
might add that nobody has ever “noticed” a place except that it was
associated with a secondary quality and that the “objects of our
perception” invariably include such qualities. Our scientific needs are
served quite simply, though: To all x, y, z, t systems of values we can
simply associate appropriate vector coordinates for the corresponding
secondary properties. Since the color at a point in spacetime does not
depend the system of coordinates chosen, we have a straight path into
the heart of both relativity and quantum theory. Moreover, “the tensor
calculus emerges as the natural framework with which to address such
matters ...” as P.M. Churchland says of neural networks. But is this
move justified on mathematical grounds? Here is Weyl again to help us
out again: "The characteristic of an n-dimensional manifold is that
each of the elements composing it (in our examples, single points,
conditions of a gas, colors, tones) may be specified by the giving of n
quantities, the "co-ordinates," which are continuous functions within
the manifold." [48] 10)
Logic Let’s
remember Weyl in regard to the mapping of colors to the points of the
projective plane: “the projective plane and the color continuum are
isomorphic with one another. Every theorem, which is correct in the one
system,_{ }is transferred unchanged to the other_{”.
With
this result in mind, let’s visit with Newman and Nagel in their classic
essay on Gödel’s proof:} How
did Gödel prove his conclusions? Up to a point, the structure of his
demonstration is modeled, as he himself noted, on the reasoning
involved in one of the logical antinomies known as the “Richard
Paradox,” first propounded by the French mathematician, Jules Richard,
in 1905...The reasoning in the Richard Paradox is evidently fallacious.
Its construction nevertheless suggests that it might be possible to
“map” (or “mirror”) meta-mathematical statements about
a sufficiently comprehensive formal system into
the
system itself. If this were possible, then metamathematical statements
about a system would be represented
by statements
within the system. Thereby
one could achieve the desirable end of getting the formal system to
speak about itself — a most valuable form of self-consciousness. (Our emphasis) The
idea of such mapping is a familiar one in mathematics. It is employed
in coordinate geometry, which translates geometric statements into
algebraic ones, so that geometric relations are mapped onto algebraic
ones. The idea is manifestly used in the construction of ordinary maps,
since the construction consists in projecting configurations on
the surface of a sphere onto a plane . . . The
basic fact which underlies all these mapping procedures is that an
abstract structure of relations embodied in one domain of objects is
exhibited to hold between “objects” in some other domain. In
consequence, deductive relations between statements about the first
domain can be established by exploring (often more conveniently and
easily) the deductive relations between statements about their
counterparts. For example, complicated geometrical relations between
surfaces in space are usually more readily studied by way of the
algebraic formulas for such surfaces. [49] Just
for the
sake of argument, suppose we interpret the elements of a formal theory T (of a very general nature) as Feigl’s
“immediately experienced qualities”. Say T
has a sufficiently rich structure so as to allow it to
frame statements about itself within itself, a la Gödel. T
will be unable to define its elements, for if it could, its elements
would not be elements. Are we similarly unable to define the elements
of our experience, the “immediately experienced qualities” in respect
of anything simpler? Is this basic fact of experience an artifact of
the fundamental logic of our brains and sense organs and of matter
generally? So
again we ask: Do the two vector manifolds always co-vary because they
are, in fact, two aspects of one manifold? It seems an altogether
reasonable question. 10)
Fields Going
a bit further, we could consider the fact that photons are routinely
described in terms of fields, as in QFT, where the photon is known as
the "exchange particle" of the electromagnetic (EM) field. Consider,
then, that the visual field is ... well, a field - and this, according
to a simple definition, such as we find in 't Hooft:
"a
field is simply a quantity defined at every point throughout some
region of space and time." [50] Or
in Dyson: "This is the characteristic mathematical property of a
classical field: it is an undefined something which exists throughout a
volume of space and which is described by sets of numbers, each set
denoting the field strength and direction at a single point in the
space." [51] On the present view, Dyson’s "undefined something" in the
visual field is just color, because, as Maxwell, Russell, and
Wittgenstein have pointed out, color is simply too simple to be
defined, and is therefore mathematically “elemental”. Such
questions raise many another in their wake — just what is color space,
e.g.? Note that we can make a natural mapping from the spectral colors
to a color sphere, where Newton’s color wheel runs around the
circumference, with black and white at the poles. Or such a mapping
could be made with red, green and blue for the axes of a unit sphere in
Hilbert space. We could then easily map those color vectors to the
photonic vectors with which they are associated, remembering that these
“physical” vectors recapitulate the mathematics of colors under vector
addition and multiplication. Then, any operation upon the photonic
vector would naturally correspond to a rotation of the color vector, in
a direct analogy with the mathematics of gauge theory and quantum
theory generally. If
such a color sphere were to "sit over" every point in 4D space-time,
that would seem to square with the facts of our experience of the
visual field and also perhaps with the mathematics of string/M-theory,
where a 6-dimensional orbifold is often pictured "sitting over" every
point in 4D spacetime (for a nice graphic illustration of this notion,
see Brian Greene's popular work The
Elegant
Universe). On
a somewhat more technical note, we might reflect on the fact that
Calabi-Yau spaces are known to be projective spaces, and to include the
SU(3) group of the standard model. So what? Well, as Weyl tells us,
colors are isomorphic to the points of the projective plane, SU(3) is
thought to govern quantum chromodynamics. The theory is called "chromo"
dynamics because quarks and gluons recapitulate the behavior of
spectral colors. Now, as ‘t Hooft, has pointed out, it is usually
thought that spectral colors and the colors of chromodynamics have
nothing to do with each other ... though no one ever says why this must
be so — it is just the sort of thing that everybody knows. Yet photons
and gluons are both gauge particles, and photons are continually
associated with spectral color. So we might defy convention and ask: Is
color a kind of gauge information? If
we were to include spectral color in physical theory, we would seem to
need additional dimensions to spacetime in order to account for the
color perceived at a point in spacetime. As is well known, there are
today various physical theories, which require increasing the number of
spacetime dimensions or variables, though on different grounds
altogether. Thus, e.g., in the original Kaluza-Klein theory, both
electromagnetism and gravity were seen to flow from a single spacetime
metric, where the number of spatial dimensions is raised from three to
four. A similar situation obtains today with string/M-theory, where all
the known forces are represented. In both K-K and M-theory the extra
dimensions are assumed to be very small because we do not “see” them.
We suggest instead that we do, in fact, “see” these other dimensions
all the time. We reason that these extra dimensions are perhaps where
the secondary properties reside. We are clearly on speculative ground,
here, but it seems at least initially plausible that what we think of
as spectral color (together with those other "elements of reality"
known to us as sound, heat and cold, etc.,) might be kinds of gauge
information — which interpretation might then address the supposed
causal inefficacy of these phenomenal properties. Of
central importance to such a research program must be the evident
differences between the usual spacetime manifold of relativity and the
manifolds of ‘colors and tones’ and so forth. Fortunately, the
mathematics of differential geometry provides us with a ready guide in
these matters. Let
us first remind ourselves of the proposal with which we began. The
three-dimensional encoding of color spaces, for example, suggests that
the information state in a color experience corresponds directly to an
information state in the brain. We might even regard the two states as
distinct aspects of a single information state, which is simultaneously
embodied in both physical processing and conscious experience. Let’s
tighten that up a bit with some help from Lockwood: Take
some
range of phenomenal qualities. Assume that these qualities can be
arranged according to some abstract n-dimensional
space, in a way that is faithful to their perceived similarities and
degrees of similarity — just as, according to Land, it is possible to
arrange the phenomenal colors in his three-dimensional color solid.
Then my Russellian proposal is that there exists, within the brain,
some physical system, the states of which can be arranged in some
n-dimensional state space ... And the two states are to be equated with
each other: the phenomenal qualities are identical with the states of
the corresponding physical system. [52] One
way of looking at our present proposal consists in considering each
perceptual state space (vision, audition, etc.,) as a projection of a
complete, n-dimensional state space that is identically phenomenal and
quantum. Remarkably, Lockwood and I arrived at this point, and by many
of the same paths, almost simultaneously, though entirely independently
— a conclusion with which Lockwood concurs (private communication). All
right, then: perceptual states are quantum states. We have been
speaking of states and spaces. We have raised the possibility that the
secondary qualities might reside in the “internal” spaces of gauge
theory, wherein the symmetries of those secondary properties might then
contribute to the QM action. 11)
Gauge Potentials and Fields So
let us attend Atiyah, who requires no introduction among mathematicians
(suffice it to say that he has made fundamental contributions to
topology and has edited a collection of essays by Fields Medal
laureates): We
shall now recall the data of a classical theory as understood by
physicists and then reinterpret them in geometrical form. Geometrically
or mechanically we can interpret this data as follows. Imagine a
structured particle, that is a particle which has a location at a point
x of R_{4} and an internal structure, or set of
states, labeled
by elements g of G. We then consider the total space P of all states of
such a particle. In general we conceive of the internal spaces G_{x}
and G_{y} for x š
y as not
being identified and so we draw the picture of P as a collection of
"fibers".[53] Here
we
regard the colors as observed “states" of the photon: In
the
absence of any external field however we consider that all G_{x}
can be identified to each other so that in addition to the vertical
lines or fibers we can also draw horizontal lines (called sections)
making the usual Cartesian type of grid. Now we can imagine an external
field imposed which has the effect of distorting the relative alignment
of the fibers so that no coherent identification is possible between
the G_{x} at different points. However we assume
that G_{x}
and G_{y} can still be identified if we choose a
path in R_{4}
from x to y. In more physical terms we imagine the particle moving from
x to y and carrying its internal space with it. Here
I imagine an interaction-free photon of constant energy carrying along
its constant or invariant color in some sort of internal space —
remembering that this kind of invariance leads us directly into the
deep waters of contemporary physical theory, where such invariances or
symmetries are understood to determine the quantum mechanical action.
Back to Atiyah: In
Minkowski space such a motion would take place along the world line of
the particle. This identification of fibers along paths is called
"parallel transport". If we now imagine two different paths joining x
to y then there is no reason for the two different parallel transports
to agree and they are assumed to differ by multiplication with a group
element, which could be viewed as a generalized "phase shift". This
phase shift is interpreted as produced by the external field. In
geometrical terms it is viewed as the total "curvature" or distortion
of the fiber bundle over the region enclosed by the two paths. [53] Here
I would point to the observational fact that a change in a photon's
color can be brought about by an "external field" such as a
gravitational field, or, by the principle of equivalence, an
acceleration field. Also, it seems to make a kind of sense to regard
such a change in color as a 'generalized phase shift'. When viewed in
respect of the total "curvature" of the fiber bundle, such a phase
shift would seem to have a natural analog in Kaluza-Klein theory, where
all interactions arise from curvature in an extended metric, in direct
analogy with gravity in general relativity. So we seem to have a clear
path toward understanding how color might couple to the equations of
relativity — i.e., via the red-shift phenomenon. Now, the prediction of
the red shift was one of three predictions offered by Einstein in
support of the general theory. I would extend the reach of that
prediction by asserting that it ought to be impossible to tell, by
judging the color of a photon, whether a shift in color to the red was
due to a gravitational source, or to a receding source. Let’s
get
back to the internal spaces, with Cao: In
this case the question of what is the relation between internal and
external geometries seems to have turned out to be quite irrelevant.
But in a deeper sense the question remains profound. What is the real
issue when we talk about the geometrization of gauge field theory? The
geometrization thesis only makes sense if the geometrical structures in
four-dimensional space-time are actually correlated to the gauge
interactions other than gravity or supergravity, or equivalently, if
they are mixed with the geometrical structures associated with extra
dimensions. [54] Here
I would point to the fact that photons, as "geometrical structures in
four-dimensional space-time" are correlated with color vectors — which
vectors we might then regard as "geometrical structures associated with
extra internal: dimensions." And, as we see, “the geometrical
structures in four-dimensional space-time” ...
“are
mixed with the geometrical structures associated with extra
dimensions”, in the manifest sense that, to each spacetime vector in
(say) our visual manifold there corresponds a color vector. A
question arises: is color phase information? That is not an issue to be
settled here, or very soon, but let's have a glance at a bit of
introductory gauge theory, just to finish up and look ahead a bit: The
extension of a global symmetry to a local gauge invariance is termed
'gauging the symmetry'...The new fields that are introduced are called
gauge boson fields, and the quanta of these fields are called gauge
particles. Thus the photon is the gauge boson of the electromagnetic
field. A natural question arises: 'when is it useful to gauge a
symmetry?'...experimental observation is the arbiter: gauging the U(1)
charge symmetry of QED forces the introduction of a gauge boson field
with exactly the properties of the experimentally observed photon
field. [55] Here
I would
simply point to the durable fact that colors, e.g.,
belong
among "the properties of the experimentally observed photon field".
However, since colors, together with the other secondary properties,
have traditionally been considered nonphysical, their evident group
properties have not been taken into consideration heretofore in regard
to the electromagnetic field: Consider
the field of the data of sense a field of universal interest and
fundamental. We are here in the domain of sights and sounds and motions
among other things...Do the colors constitute a group?...Let us pass
from colors to figures or shapes to figures or shapes, I mean, of
physical or material objects rocks, chairs, trees, animals and the like
as known to sense perception ... And what of sounds, sensations of
sound? Are sounds combinable? Is the result always a sound or is it
sometimes silence? If we agree to regard silence as a species of sound
— as the zero of sound — has the system of sounds the property of a
group? [56] Recall
that the addition of color vectors reliably depends upon the phases of
their concomitant state vectors. Repairing the rift between primary and
secondary, and so between mind and body, would seem to require a larger
group of symmetries for the EM field and for nature as a whole.
Happily, because the secondary properties are given to us in
observation, in predictable conjunction with photons in characteristic (eigen) states,
the determination of the complete symmetry group is open to our
inspection - a fact of some importance given Weinberg’s remark that “it
is increasingly clear that the symmetry group of nature is the deepest
thing that we understand about nature today.” To
sum up and look ahead a little bit, then, we started out with Chalmers
reasoning that certain information states in the brain embody both
“physical processing and conscious experience”. Along with Feigl, we
have considered that the relevant states are the “immediately
experienced qualities, or their configurations in the various
phenomenal fields” which embody the concomitant quantum fields as well.
In so doing, we have looked at various points of contact between the
mathematics of quantum fields and that of the visual field. However,
since the secondary qualities are not mentioned in the formalisms of
current quantum theory, and since these properties appear to be
elemental, we have looked at the foundations of physical theory for
places where additional variables or dimensions might be accommodated,
viz., hidden variables theory, gauge theory, and string/M-theory. Since
the first of these alternatives is weighed down with an excess of
rhetorical baggage, it may be most expeditious to pursue the latter two
possibilities, which offer the further advantage of being highly
developed mathematically. With these points in mind, let us finish up
by recalling Wittgenstein on
color space: "a
speck in the visual field, though it need not be red must have some
color; it is, so to speak, surrounded by color-space. Notes must have
some pitch, objects of the sense of touch some degree of hardness, and
so on". It
is as though each speck in the visual field is tangent to color space —
as though a color sphere “sits over” each spacetime coordinate of the
visual field, in direct analogy with string/M-theory and Kaluza-Klein
theory. Now let us reflect on the following, where TpX is the tangent space of X at p.
and
where v is a color vector living on a color sphere associated with a
photonic state vector moving along the curve C: First, let X be a real differentiable manifold of real dimension n, and let v ∈ TpX. Assuming
that X is equipped with a metric g and the associated Levi-Civita
connection G, we can imagine parallel transporting v along a curve C in
X which begins and ends at p. After the journey around the curve, the
vector v will generally not return to its original orientation in TpX.
Rather, if X is not at, v will return to p pointing in another
direction, say v^{’} (since we are using the
Levi-Civita
connection for parallel transport, the length of v will not change
during this process) If X is orientable, the vectors v and v^{’}
will be related by an SO(n) transformation AC, where the subscript
reminds us of the curve we have moved around. That is v'
= AC_{v} : (2.44) Now
consider all possible closed curves in X which pass through p, and
repeat the above procedure. This will yield a collection of SO(n)
matrices AC1 ; AC2 ; AC3 ; :::, one for each curve. Notice that if we
traverse a curve C which is the curve Ci followed
by the curve Cj , the associated matrix will be ACj ACi and that if we
traverse the curve Cj in reverse, the associated matrix will be A^{-1}∈Cj
. Thus,
the collection of matrices generated in this manner form a group —
namely, some subgroup of SO(n). Let us now take this one step further
by following the same procedure at all points p on X. Similar reasoning
to that just used ensures that this collection of matrices also forms a
group.[57] Well,
when Brian Greene wrote the foregoing, he was explaining Calabi-Yau
spaces with respect to string theory. But if I am not mistaken, his
remarks above apply equally well to color vectors, and so, echoing
Weyl, it seems like it might be useful to develop this geometry a
little further.
References: [1]
David J.
Chalmers, "Puzzle of Conscious Experience", Scientific
American, 12/95. [2]
"Mind-body, not a pseudo problem" Herbert Feigl.
The
Mind-Brain Identity Theory. Borst, CV, ed. New York, NY: St.
Martin's Press, 1970. [3]
Bohm, D.
and Hiley, BJ. The Undivided Universe,
pp. 386-6 [4]
Bohm,
David, Quantum Theory. Englewood
Cliffs, NJ:
Prentice-Hall, Inc., 1951. [5]
Dyson,
Freeman J., "Field Theory", pp. 58-60, Scientific American, 188: 1953. [6]
Salam,
Abdus, Unification of Fundamental Forces.
Cambridge,
1990. [7]
Pribram,
Brain and Perception.
Hillsdale, NJ: Lawrence Erlbaum,
1991. [8]
Umezawa,
Hiroomi. Advanced Field Theory. New
York, NY: American
Institute of Physics, 1993. [9]
Wigner,
“Physics and the Explanation of Life,” in Foundations
of
Physics, vol. 1, 1970, pp. 34-45. [10]
Lockwood, Michael. Mind, Brain and the Quantum.
Cambridge, MA: Basil Blackwell Ltd., 1989. [11]
Hawking, SW, and Ellis, GFR. The large-scale
structure of
space-time, Cambridge, 1973. [12]
Brown,
H and Harré, R. Philosophical Foundations of
Quantum Field
Theory. Oxford: Oxford University Press, 1988. [13]
Churchland, P M A Neurocomputational
Perspective.
Cambridge, MA: MIT Press, 1989. [14]
http://www.ucc.uconn.edu/~wwwphil/cunyall.html [15]
Feynman, Richard, and Weinberg, Steven. Elementary
particles and the laws of physics. New York, NY: Cambridge
University Press, 1987. [16]
Weyl,
H. Mind and Nature, pp. 8-9,
University of
Pennsylvania Press, 1934. [17]
Weyl,
ibid., p. 10. [18]
Byron,
Frederick W, Jr., and Fuller, Robert W. Mathematics
of
Classical and Quantum Physics. Dover, NY, 1970. [19]
Dirac,
PAM. The Principles of Quantum Mechanics.
Oxford, 1958. [20]
Hughes,
RIG. The Structure and Interpretation of
Quantum Mechanics.
Cambridge, MA: Harvard University Press, 1989. [21]
Wittgenstein, Ludwig. Tractatus
Logico-Philosophicus.
Atlantic Highlands, NJ: Humanities Press, 1974. [22]
Poincaré, Henri. “Space and Geometry” in The
Foundations
of Science. New York, NY: The Science Press, 1913, pp 66-80. [23]
Weyl,
ibid., pp. 18-20. [24]
Kline,
Morris, “Projective Geometry”, Scientific
American,
1955. [25]
Holland, Peter R. The Quantum Theory of Motion.
Cambridge University Press, 1993. [26]
Schrödinger, Erwin. Mind and Nature.
Cambridge
University Press, 1959. [27]
Weyl,
ibid., p. 24 [28]
Einstein, A, “On Bertrand Russell’s Theory of Knowledge”, Albert
Einstein Philosopher-Scientist, ed., Schilpp, LaSalle, Open
Court,
1970. [29]
Clark,
Austen. Sensory Qualities.
Clarendon Library of Logic
and Philosophy, Oxford. 1993. [30]
Hume,
David, A Treatise of Human Nature,
Oxford, 1968. [31]
Mach,
“Analysis of Sensations” excerpted in Body
and Mind,
ed., Vesey, GNA, Allen & Unwin, 1970. [32]
Weyl,
ibid., pp. 75-6. [33]
Weinberg, “Towards the final laws of physics”, Elementary
Particles and the Laws of Physics, Cambridge, 1987. [34]
Lopuszanski, Introduction to Symmetry
& Supersymmetry
in Quantum Field Theory: Singapore, World Scientific, 1991. [35]
Einstein, A, Podolsky, B, and Rosen, N. “Can Quantum-Mechanical
Description of Physical Reality Be Considered Complete?” Physical
Review, Vol. 47, 777 (1935). [36]
Holland, Peter R. The Quantum Theory of Motion.,
Cambridge University Press, 1993. [37]
Holland, ibid. [38]
Wheeler
& Tegmark, “100 Years of Quantum Mysteries”, Scientific
American, 2/2001. [39]
Newton,
Isaac. Opticks, Dover, 1979. [40]
Riemann, D. “On the Hypotheses Which Lie at the Bases of Geometry”, Nature, Vol. VIII, 183-4, pp. 14-17. [41]
Weyl,
H. Space Time Matter, p 3, Dover
1922. [42]
Davies,
PCW, ed., Superstrings: A Theory of
Everything?
Cambridge, 1986. [43]
Davies,
PCW, ibid. [44]
Peter
G. Bergmann, “The Quest for Unity: General Relativity and Unitary Field
Theory” Introduction to the Theory of
Relativity,
Prentice-Hall, New York, 1943. [45]
Kaluza,
T. "On the Unity Problem of Physics" in Modern
Kaluza-Klein Theories, Addison-Wesley,
Menlo
Park, 1987. [46]
Cao,
"Gauge Theory". Philosophical Foundations of
Quantum Field
Theory. Brown, H and Harré, R. eds., Oxford University Press,
1988. [47]
Minkowski, “Space and Time”, The Principle of
Relativity,
ed., HA Lorentz, Methuen & Co., London, 1923. [48]
Weyl, Space Time Matter, p. 84,
Dover, 1922. [49]
Nagel
and Newman, “Goedel's Proof”, World of
Mathematics,
vol. III, ed., James Newman, Simon & Schuster, 1956. [50]
Gerard
't Hooft, "Gauge Theories of the Forces between Elementary Particles",
p. 81, Scientific
American,
6/80. [51]
Dyson,
Freeman J., "Field Theory", pp. 58-60, Scientific
American,
188: 1953. [52]
Lockwood, ibid. [53]
Atiyah,
MF. Geometry of Yang-Mills Fields.
Pisa, Italy:
Accademia Nazionale Dei Lincei Scuola Normale Superiore, 1979. [54]
Cao,
ibid. [55]
Guidry,
M. Gauge Field Theories. New York,
NY: Wiley and Sons,
1991. [56]
Keyser,
C. “The Group Concept”, Mathematical
Philosophy, New
York, 1922. [57]
Greene,
Brian R. String Theory on Calabi-Yau Manifolds, hep-th/9702155 23 Feb |
||