Who sees variety and not the Unity,
wanders from death to death. ~Upanishads
't Hooft, Saunders, Dyson, Maxwell, Pribram, Umezawa, Chalmers, Feynman, Green, Bohm, Hume, EPR
There is nothing else except these fields: the
whole of the material universe is built of them. ~Dyson
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.
Fourier's theorem is probably the most far-reaching
principle of mathematical physics. ~Feynman
Many of the
foundations of wave mechanics are based on the analyses and equations
derived for the theory of acoustics in his book The Theory of Sound.
Erwin Schrödinger, a pioneer in quantum mechanics, studied this book
and was familiar with the perturbation methods it describes.
A complex of coefficients of this type is comparable with a matrix such as occurs in linear algebra. ~Heisenberg
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 [...]
have been many models based on quantum theories, but many of them are
rather philosophiically 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.
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. ~ Bohm
I feel very strongly that the stage physics has reached at the present day is not the final stage. It is just one stage in the evolution of our picture of nature, and we should expect this process of evolution to continue in the future, as biological evolution continues into the future. The present stage of physical theory is merely a steppingstone toward the better stages we shall have in the future. One can be quite sure that there will be better stages simply because of the difficulties that occur in the physics of today.
The fundamental principle of that philosophy is the opinion concerning colours, 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, colours, 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. These primary qualities are extension and solidity, with their different mixtures and modifications; figure, motion, gravity and cohesion. The generation, encrease, decay and corruption of animals and vegetables, are nothing but changes of figure and motion; as also the operations of all bodies on each other; of fire, of light, water, air, earth, and of all the elements and powers of nature [...]
Thus there is a direct and total opposition betwixt our reason and senses [...] When we reason from cause and effect, we conclude, that neither colour, 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.
The mathematical machinery of quantum mechanics
became that of spectral analysis... ~Steen
A theory that yields "maybe" as an answer should
be recognized as an inaccurate theory. ~ 't Hooft
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.
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.
Thus the colors with their various qualities and intensities fulfill
Surprisingly, the Langlands dual group also appears in quantum physics in what looks like an entirely different context; namely, the electro-magnetic duality. Looking at the Maxwell equations describing the classical electromagnetism, one quickly notices that they are invariant under the exchange of the electric and magnetic fields. It is natural to ask whether this duality exists at the quantum level.
This is a classical system that exhibits behavior that people previously thought was exclusive to the quantum realm, and we can say why,” said John Bush, a professor of applied mathematics at MIT who has led several recent bouncing-droplet experiments. “The more things we understand and can provide a physical rationale for, the more difficult it will be to defend the ‘quantum mechanics is magic’ perspective.
It takes an extraordinary intelligence
to contemplate the obvious. ~Whitehead
A field is simply a quantity defined at every point
throughout some region of space and time. ~'t Hooft
Physicists talk about two kinds of fields: classical fields and quantum fields. Actually, we believe that all fields in nature are quantum fields. A classical field is just a special large-scale manifestation of a quantum field.
But since classical fields were discovered first and are easier to understand, it is necessary to say what we mean by a classical field first, and go on to talk about quantum fields later.
A classical field is a kind of tension or stress which can exist in empty space in the absence of matter. It reveals itself by producing forces, which act on any material objects which happen to lie in the space the field occupies.
In order to describe completely the state of the fields in a given region of space, it is necessary to specify the strength and the direction of both the electric and the magnetic fields at every point of the region separately. 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.
chemical binding is electromagnetic in origin,
Now the immediate fact which psychology, the science of mind, has to study is also the most general fact. It is the fact that in each of us, when awake (and often when asleep), some kind of consciousness is always going on. There is a stream, a succession of states, or waves, or fields (or of whatever you please to call them), of knowledge, of feeling, of desire, of deliberation, etc., that constantly pass and repass, and that constitute our inner life. The existence of this stream is the primal fact, the nature and origin of it form the essential problem, of our science. So far as we class the states or fields of consciousness, write down their several natures, analyze their contents into elements, or trace their habits of succession, we are on the descriptive or analytic level. So far as we ask where they come from or why they are just what they are, we are on the explanatory level.
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.
A projection matrix P is an n x n square matrix that gives a vector space projection from Rn to a subspace W. The columns of P are the projections of the standard basis vectors, and W is the image of P.
conjecture is thus that M-theory formulated in the infinite momentum
frame is exactly equivalent to the N → ∞ limit of the supersymmetric
quantum mechanics described by the Hamiltonian. The calculation of any
physical quantity in M-theory can be reduced to a calculation in matrix
quantum mechanics followed by an extrapolation to large N.
Susskind et al.
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.
In ordinary Minkowski space the electro- magnetic force is described by a 2-form (skew- symmetric 2-tensor) . In this notation, Maxwell’s equations in vacuo are
where now is defined using the Lorentz metric in .
Formally they are the same as Hodge equations for forms of degree 2 on a 4-dimensional Riemannian manifold, but here the space is ordinary Minkowski space, not the 4-dimensional Euclidean space. From this it appears that Maxwell’s equations, which unified electricity and magnetism, also encode a duality between electricity and magnetism in the sense that the operator interchanges both aspects. Physically, this is a very fundamental fact of the universe.
A lot of modern physics is concerned about gauge theory. This is what physicists use to describe elementary particles. In some naive sense it is just a (non-abelian, non-linear) matrix generalization of Maxwell’s theory.
A Yang-Mills field F is the curvature of a connec- tion on a fibre bundle (F may be thought as a as a 2-form with matrix coefficients, or as a matrix of 2-forms). Then the Yang-Mills equations take the same form again:
These are now matrix equations. When defined on Lorentz manifolds, they are the fundamental equations that physicists use for elementary & particle physics. They can also be defined on a Riemannian 4-manifold, and this leads to geometry, as in Donaldson’s celebrated theory, one of the most exciting developments in the last quarter of the 20th century. In particular it produced new invariants of 4-manifolds.
It is important to ephasise however that, while can distinguish these different lines of work — involving very different mathematical techniques — the activities were closely related. Thus one notable feature of the impact of Yang-Mills theory within mathematics has been to increase the unity of the subject, throwing bridges between mathematical areas (e.g. PDE theory, vector bundles over complex projective space) which might have been seen before as having little connection.
The Langlands program has, along these lines, something called the geometric Langlands program, which replaces the number fields by Riemann surfaces. It is a very interesting theory which is much easier than the number field case, but not trivial and still quite big. It is developed by using the theory of vector bundles on Riemann surfaces.
The three-dimensional encoding of colors, 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.
Repeated time and again with unimaginably more sophisticated
and sensitive apparatus than Young's, the double-slit experiment
encapsulates, said the physicist Richard Feynman, the "heart of
quantum mechanics," its "only mystery."
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.
Einstein and Bohr had a debate.
There is now in my opinion no entirely satisfactory
interpretation ofquantum mechanics. ~Weinberg