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Action  |
Roughly speaking, force is the space derivative of energy and the time derivative of momentum. You can take one more step up the ladder: energy and momentum are both derivatives of action: energy is its time derivative, momentum its space derivative. Wilczek  It is a
most beautiful and awe-inspiring fact that all the fundamental laws of
Classical Physics can be understood in terms of one mathematical
construct called the Action. It
yields the classical equations of motion, and analysis of its
invariances leads to quantities conserved in the course of the
classical motion. In addition, as Dirac and Feynman have shown, the
Action acquires its full importance in Quantum Physics.
It is increasingly clear
that the symmetry
group
of nature is the deepest thing that we understand about nature today.
 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 come 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.
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 [...] it was found possible to
account for the atomic stability, as well as for the empirical laws
governing the spectra of the elements, by assuming that any reaction of
the atom resulting in a change of its energy involved a complete
transition between two so-called stationary quantum states and that, in
particular, the spectra were emitted by a step-like process in which
each transition is accompanied by the emission of a monochromatic light quantum of
an energy just equal to that of an Einstein photon.
Bohr
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Audition |
 The fact that the formalism describing the brain microprocess is identical with the physical microprocess allows two interpretations: (a) The neural microprocess is in fact based on relations among microphysical quantum events, and (b) that the laws describing quantum physics are applicable to certain macrophysical interactions when these attain some special characteristics.” Pribram
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?
Keyser
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Brain |
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.
Umezawa
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Cerebellum |
 In Academic
Research, Pellionisz, as professor
of New York University was the originator of a pioneering
Information Geometry approach to Neural Nets, Tensor Network Theory. TNT
explains the function of 1/4 of the brain (the cerebellum) in
terms of tensor analysis, the intrinsic
mathematical language of Biological Neural Nets.
Pellionisz
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Cerebrum |
 The cerebrum makes up about 85 per cent of the weight of the human brain. A large groove called the longitudinal fissure divides the cerebrum into halves called the left cerebral hemisphere and the right cerebral hemisphere. The hemispheres are connected by bundles of nerve fibres, the largest of which is the corpus callosum. |
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Color |
It
seems useful to me to develop a little more precisely the "geometry"
valid in the two-dimensional manifold of perceived colors. For one can
do mathematics also in the domain of these colors. The
fundamental operation which can be performed upon them is mixing: one
lets colored lights combine with one another in space [...]
Weyl A color is a physical
object a soon as we consider its dependence, for instance, upon its
luminous source, upon temperatures, and so forth. Mach
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 Computer: Quantum |
Now, researchers have developed a new, yet less exotic computing method that may be as good as quantum computing for certain tasks, such as searching databases. The method relies entirely on classical physics, say Ian Walmsley and his colleagues of the University of Rochester in New York. To convert their ideas into hardware, the Rochester scientists have built an optical device and successfully demonstrated the method. The group reported its results at the Lasers and Electro-Optics/Quantum Electronics and Laser Science conference in Baltimore last week. Researchers expect
quantum processors to work incredibly fast thanks in part to particles'
wavelike interactions, including interference. The processors would
take advantage of another, stranger effect known as entanglement, in
which two or more particles share one quantum state.
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Cortex |
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Dendrite |
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Dualism |
In dualism, ‘mind’ is
contrasted with ‘body’, but at different times, different aspects of
the mind have been the centre of attention. In the classical and
mediaeval periods, it was the intellect that was thought to be most
obviously resistant to a materialistic account: from Descartes on, the
main stumbling block to materialist monism was supposed to be
‘consciousness,’ of which phenomenal consciousness or sensation came to
be considered as the paradigm instance.
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Duality |
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EM |
 Fundamental electromagnetic interactions occur between any two particles that have electric charge. These interactions involve the exchange or production of photons. Thus, photons are the carrier particles of electromagnetic interactions. |
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Energy |
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EPR |
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.EPR
Einstein
 Thus
"this is red,"
"this is earlier than that," are atomic propositions.
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Eye |
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Field |
A field is simply a quantity defined at every point throughout some region of space and time.
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Form |
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Fractal |
 Does neural form follow quantum function? |
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Gauge |
As far as gravity is concerned, Einstein’s General Relativity is a beautiful and complete theory. But as Einstein realized it has to be extended to account for other physical forces, the most notable being electro-magnetism. It is perhaps no accident that the first and most significant step in this direction was taken by a mathematician – Hermann Weyl. He showed that, by adding a fifth dimension, electromagnetism could also be interpreted as curvature. His idea was that the size of a particle could alter as it passed through an electro-magnetic field. In analogy with railways it was called a gauge theory, and this name has stuck through subsequent evolutions of the theory. Unfortunately for Weyl, Einstein immediately objected on physical grounds that this would have meant different atoms of, say hydrogen, would have different sizes depending on their past history, in contradiction with observation. Given this devastating critique, it is remarkable but fortunate that Weyl’s paper was still published, with Einstein’s objection as an appendix. Clearly the beauty of the idea attracted the editor, despite the fatal flaw. In fact, beauty often wins such contests, because with the advent of quantum mechanics, with its complex wave functions, it was pointed out by Kaluza and Klein that Weyl’s gauge theory could be salvaged if one interpreted the variable as a phase rather than a length. A pure phase shift by itself is not physically observable and so Weyl’s theory avoids the Einstein objection. Atiyah |
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Gravity |
So the local gauge symmetry also requires the introduction of gauge potentials, which are responsible for the gauge interactions, to connect internal directions at different space-time points. We also find that the role the gauge potentials play in fiber-bundle space in gauge theory is exactly same as the role the affine connection plays in curved space-time in general relativity.  |
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Group |
A group 1. Closure: If 2. Associativity: The
defined multiplication is associative, i.e., for all 3. Identity: There is an identity element 4. Inverse: There must be
an inverse (aka reciprocal) of each element. Therefore, for each
element
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Hidden Variables |
Well, obviously the extra dimensions have to be different somehow because otherwise we would notice them. Green
Now it may be asked why these hidden variables should have so long remained undetected.
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Information |
"One day I had a drink with some machine-learning researchers, and we suddenly said, 'Oh, it's not noise,' because noise implies something's wrong," says Pouget. "We started to realize then that what looked like noise may actually be the brain's way of running at optimal performance." Bayesian computing can be done most efficiently when data is formatted in what's called "Poisson distribution." And the neural noise, Pouget noticed, looked suspiciously like this optimal distribution.
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Invariants |
 It is a little hard to
understand the significance of Klein's contributions to geometry. This
is not because it is strange to us today, quite the reverse, it has
become so much a part of our present mathematical thinking that it is
hard for us to realise the novelty of his results and also the fact
that they were not universally accepted by all his contemporaries. [...]
During his time at Göttingen in 1871 Klein made major discoveries regarding geometry. He published two papers On the So-called Non-Euclidean Geometry in which he showed that it was possible to consider euclidean geometry and non-euclidean geometry as special cases a projective surface with a specific conic section adjoined. This had the remarkable corollary that non-euclidean geometry was consistent if and only if euclidean geometry was consistent. The fact that non- euclidean geometry was at the time still a controversial topic now vanished. Its status was put on an identical footing to euclidean geometry. Cayley never accepted Klein's ideas, believing his arguments to be circular.  Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872), profoundly influenced mathematical development. This was written for the occasion of Klein's inaugural address when he was appointed professor at Erlangen in 1872 although it was not actually the speech he gave on that occasion. The Erlanger Programm gave a unified approach to geometry which is now the standard accepted view. |
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Matrix |
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Matter |
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M-theory |
M theory is a name for a more
unified theory that has the different string theories, as we
know them, as limits, and which also can reduce, under appropriate
conditions, to eleven-dimensional supergravity. There's this picture
that we all have to draw where different string theories are limits of
this M theory, where M stands for Magic, Mystery or Matrix, but it also
sometimes is seen as standing for Murky, because the truth about M
theory is Murky.
Witten
While a proper understanding of M-theory still eludes us, much is now known about it. In particular the various geometric results that have emerged from string theory become related in interesting but mysterious ‘dualities’ whose real meaning has yet to be discovered. Atiyah
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 [...] the projective plane and the color continuum are isomorphic with one another. Weyl
The principle of duality in projective geometry states that
we can interchange point and line in a theorem about figures lying in
one plane and obtain a meaningful statement. Moreover, the new or dual
statement will itself be a theorem—that is, it can be proven. On the
basis of what has been presented here we cannot see why this must
always be the case for the dual statement. However, it is possible to
show by one proof that every rephrasing of a theorem of projective
geometry in accordance with the principle of duality must be a theorem.
This principle is a remarkable characteristic of projective geometry.
It reveals the symmetry in the roles that point and line play in the
structure of that geometry. Kline
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Monism |
Mass and energy are both but different manifestations of the same thing — a somewhat unfamiliar conception for the average mind. Einstein
The stuff of which the world of our experience is composed is, in my belief, neither mind nor matter, but something more primitive than either. Both mind and matter seem to be composite, and the stuff of which they are compounded lies in a sense between the two, in a sense above them both, like a common ancestor. Bertrand Russell
Most versions of neutral
monism are versions of noneliminativist reductionism. Mental and
physical phenomena are real but reducible to/constructible from the
underlying neutral level. It differs from other versions of
reductionism—be they materialistic or mentalistic, eliminative or
noneliminative—by insisting on the neutrality of the basis. And its
reductionism sets it apart from certain versions of nonreductive
theories—emergentism and the dual aspect theory come to mind—with which
it is sometimes compared or identified.
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Neural Nets |
 Artificial Neural Net (ANN) NNs with Java |
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Neuron |
 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. |
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Neuroscience |
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Optics |
In quantum
mechanics, the essential difference is that the equations of motion of
a particle are replaced by the SchrÖdinger equation for a wave. This SchrÖdinger equation
is obtained from a canonical formalism, which cannot be expressed in
terms of the fields alone, but which also requires the potentials.
Indeed, the potentials play a role, in SchrÖdinger's equation,
analogous to that of the index of
refraction in optics.
Aharanov,
Bohm
 Least action |
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Parallelism |
The Proposed Generalization The effectiveness of quantum, as compared to digital algorithms, [...] therefore suggests that spin, as the continuous spectrum of values between zero and one, with the alternative interpretation of a weighting function appropriate to neural nets, should become the Cybernetic Machine Group's principal focus for 1999. For the behaviour of spin relative to a reference spin, as in quantum entanglement, implies quantum parallelism, i.e., a superposition of all weighting possibilities simultaneously, [thus] generalizing the concept of the artificial neural net. It can [therefore] be postulated, that if such quantum neural network models [...] can be devised, (providing an understanding of their actual physics), then the key to new technology, by means of which NP complete problems can be solved, will be to hand. The Evidence for such a Generalization Such a generalization of a neural network, is, in principle, as Perus1 has shown, a highly valid concept, since the two formalisms can be set down in identical ways so as to express their properties, except that the neural net formalism concerns real quantities, while the quantum systems formalism concerns complex quantities. Weights, taking values from 0 to 1 — the key to understanding traditional neural nets — therefore become complex quantities, expressible through unit vectors (spins) in terms of phase. Wave properties and considerations of phase, could therefore contribute additional structure and understanding, both as to how neural net parallelism works, and to an explanation of the basis of the technology by means of which this can be achieved. One can then ask, do such considerations provide a better explanation of actual neuron dynamics and morphology, and if so, attempt experimental validation advancing biological understanding. |
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Particle |
 After all, our very definition of a particle or metastable nuclear state is based on its classification as the carrier of a definite representation of the Poincaré group [...]  |
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Photon |
All the
fifty years of conscious brooding have brought me no closer to the
answer to the question, "What are light quanta?" Of course today every
rascal thinks he knows the answer, but he is deluding himself.
Einstein
The question now is, how does it really work? What machinery is actually producing this thing? Nobody knows any machinery. Nobody can give you a deeper explanation of this phenomenon than I have given [...] 
Now it may be asked why these hidden variables should have so long remained undetected. Bohm
 The aspects
of things that are most important for us are hidden because of their
simplicity and familiarity. Wittgenstein |
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Projection |
 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 [...]
the projective plane and the color continuum are isomorphic with one
another. Every theorem which is correct in the one system S1
is transferred unchanged to the other S2.
A science can never determine its subject matter except up to an
isomorphic representation. The idea of isomorphism indicates the
self-understood, insurmountable barrier of knowledge. It follows that
toward the "nature" of its objects science maintains complete
indifference. This for example what distinguishes the colors from the
points of the projective plane one can only know in immediate alive
intuition [...] Weyl
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Qualia |
Feelings and experiences vary widely. For example, I run my fingers over sandpaper, smell a skunk, feel a sharp pain in my finger, seem to see bright purple, become extremely angry. In each of these cases, I am the subject of a mental state with a very distinctive subjective character. There is something it is like for me to undergo each state, some phenomenology that it has. Philosophers often use the term ‘qualia’ (singular ‘quale’) to refer to the introspectively accessible, phenomenal aspects of our mental lives. |
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Qualities |
These I call original or primary qualities of the body, which I think we may observe to produce simple ideas in us, viz., solidity, extension, figure, motion or rest, and number. Secondly, such qualities which in truth are nothing in the objects themselves, but powers to produce various sensations in us by their primary qualities, i.e. by the bulk, figure, texture, and motion of their insensible parts, as colour, sounds, tastes, etc., these I call secondary qualities. Locke
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QED |
I
would like to again impress you with the vast range of phenomena that
the theory of quantum electrodynamics describes: It's easier to say it
backwards: the theory describes all the phenomena of the physical world
except the gravitational effect ... and radioactive phenomena, which
involve nuclei shifting in their energy levels. So if we leave out
gravity and radioactivity (more properly, nuclear physics) what have we
got left? Gasoline burning in automobiles, foam and bubbles, the
hardness of salt or copper, the stiffness of steel. In fact, biologists
are trying to interpret as much as they can about life in terms of
chemistry, and as I already explained, the theory behind chemistry is
quantum electrodynamics.
Feynman
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QFT |
In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields. In a relativistic theory the wave function is a functional of these fields, not a function of particle coordinates. Quantum field theory hence led to a more unified view of nature than the old dualistic interpretation in terms of both fields and particles. |
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QM |
In 1905 Einstein examined the photoelectric effect. The photoelectric effect is the release of electrons from certain metals or semiconductors by the action of light. The electromagnetic theory of light gives results at odds with experimental evidence. Einstein proposed a quantum theory of light to solve the difficulty and then he realised that Planck's theory made implicit use of the light quantum hypothesis. By 1906 Einstein had correctly guessed that energy changes occur in a quantum material oscillator in changes in jumps which are multiples of  v
where
is Planck's reduced constant and v
is the frequency. Einstein received the 1921 Nobel
Prize for Physics, in 1922, for this work on the photoelectric effect. |
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QZE |
The very nature of
quantum physics is counterintuitive to conventional thinking. Among the
many bizarre characteristics is the quantum Zeno paradox, an odd
mathematical result that is being debated to this day. Assuming an
unstable quantum state, intuition would dictate that eventually, the
system will irreversibly decay in certain amount of time, defined as
the Zeno time. However if the system is measured in a period shorter
than the Zeno time, then the wave function of the system will
repeatedly collapse before decay. In effect, constant measurements of
the system will actually prevent its collapse! Even more mysterious, if
the time interval between measurements is longer than the Zeno time,
the decay rate of the system will increase, leading to what is termed
the anti-Zeno effect.
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Retina |
 [All] chemical binding is
electromagnetic in origin, and so are all
phenomena of nerve impulses.
Salam
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Sentience |
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Space-time |
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Symmetry |
It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today. Weinberg  The two
great events in twentieth
century physics are the rise of relativity theory
and
of quantum mechanics. Is there also some connection
between
quantum mechanics and symmetry? Yes indeed. Symmetry
plays
a great role in ordering the atomic and molecular
spectra,
for the understanding of which the principles of
quantum
mechanics provide the key.
Weyl
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Synapse |
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Tensor |
Tensors,
defined mathematically, are simply arrays of numbers, or functions,
that transform according to certain rules under a change of
coordinates. In physics, tensors characterize the properties of a
physical system, as is best illustrated by giving some examples
(below).
A tensor may be defined at a single point or collection of isolated points of space (or space-time), or it may be defined over a continuum of points. In the latter case, the elements of the tensor are functions of position and the tensor forms what is called a tensor field. This just means that the tensor is defined at every point within a region of space (or space-time), rather than just at a point, or collection of isolated points. |
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Vector |
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: Now suppose another color Y is made from the same three colors: 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: Feynman
 A field is simply a quantity defined at every point throughout some region of space and time.('t Hooft) |
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Vision |
 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.
Weyl |
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Wave |
 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. 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.
Dirac
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is a finite
or infinite set of elements together with a 
, 
, 
, ... with
binary operation between 
and 
denoted 
form a
group if 
and 
are two
elements in 
, then the
product 
is also in 
. 
, 
. 
(aka 1, 
, or 
) such that 
for every
element 
. 
of 
, the set
contain an element 
such that 
. 
