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Notes

[Sunday 23 November 2008 - Saturday 29 November 2008 ]

[Notebook: DB 64 Gravitation]

[page 167]

Sunday 23 November 2008

The alphabet of the Universe is the number of different classes of fundamental particles that it contains. The physical layer of the Universe comprises the alphabet plus the spacetime location of each particle of each type. We can have many electrons because they are in many different places How many space-time distinct electrons are there?

. . .

The fundamental element of quantum mechanical configuration space s the state, represented by a state vector in Hilbert space. From an abstract point of view all quantum states are identical, differentiated only by their state vectors.

Quantum states represent possibilities which are realized when a state is 'executed' by energy. Such a realization of a possible state is a particle. Just as quantum states may have any measure of complexity so particles may rangfe from the fundamental particles studied by physics like electrons and photons to people, planets and beyond.

[page 167]

Experience shows that fundamental particles fall into two distinct classes known as fermions and bosons. The difference between these classes appears when the particle interact, Many basons may occupy one quantum state, but only one fermion. This latter property was first noticed by Paul and is called the Pauli exclusion principle Pauli exclusion principle - Wikipedia

How to get the best out of oneself - get oneself in a position where the only way out is the desired advance, like telling everyone that you wil climb Mount Everest without oxygen. Now after nearly 64 years I have the settled ambition of developing and propagating a religious outlook which, given time and development (perhaps centuries) will underwrite peace on earth and the conditions for human survival and even heaven on earth, at least until the sun dies. An extreme ambition, but one that must be embraced of the planet is to become aesthetically pleasing to me, without all the horrendous evils that we and the planet are currently heir to.

Surowiecki NY 12/5/08 p 48: 'Corporations hope that the right concept will turn things around overnight. This is what you might call the crash-diet approach: starving your self for a few days and you will be thin for life. The Toyota approach is more like a regular sustained diet -- less immediately dramatic but, as everyone knows, much harder to sustain, In the nineteen nineties a McKinsey study of companies that had put quality improvement programs in place found that two thirds abandoned them as failures. Toyota's innovative methods may seem mundane, but their sheer relentlessness defeats many companies. That's why Toyota can afford to hide in plain sight: it knows the system is easy to understand but hard to follow.

Gladwell New Yorker re Nathan Myhrvold Malcolm Gladwell

In a system with one space and one time dimension, a rotation can be changed as changing direction along the space dimension from moving + to moving -. Electrons have only two spin states which point in different directions along a line which may have any orientation in 3-space. While an electron can change its spin direction, a photon is created with a certain spin and that cannot be changed without annihilating the photon, For a photon existing outside time the spin state is fixed. An electron, existing in spacetime, has fixed spin magnitude but variable spin direction.

Monday 24 November 2008

Spin - state occupancy theorem:
Boson - many quanta ('particles') per state
fermion - maximum one particle per state

Why?

Naive: these are two possibilities, and the Universe uses all possibilities, therefore.

Quantum mechanical: Boson wavefunctions interfere with a positive sign and so reinforce one another; fermion interfere with a negative sign and cancel.

Topological: it takes a 4 pi turn to bring an electron back to its original state - the plate dancer analogy. Transposing particles is equivalent to turning them

[page 170]

Special relativity, CPT, boson, fermion can all exist in 2D spacetime with the metric 1, -1 and the invariant interval 0 = ds2 = x2 - t2 ? Streater & Wightman

TomonagaTomonaga

Tomonaga 113 Spinors

Identity of particles = identity of processes = identical software. All identical particles are different instances of the same process.

Commute = 'simultaneously observable', ie the observation of one does not perturb the other -- they are not causally connected = (they may be) spacelike separated.

C = antiparticles
P = mirror reflection
T = time reflection

In time space we can imagine two time directions - Feynman sees an electron moving forward in time as a positron moving backwards, In 2D Minkowski space we can have P, T ans so C. Rotation in two space has only two points represented by any one of P, T or C, all of which are binary choices.

Bosons do not see this duality, fermions do? Bosons are the layer below fermions in the network model?

[page 171]

The 2D Universe can accommodate particle, anti-particle, fermion boson, spin +1. Spin -1, spin 1/2, spin -1/2 but still has no spatial extension it is equivalent to 2 bits, four states, This 2D system is in network terms the 'user' of the 1D system.

The mathematical foundations of a stable society. The physical world is a paradigm for the social world and all the other worlds. A world is a bounded layer in the symmetric network.

A network is a distributed set of memories able to communicate with and therefore change one another. You are subtly different after reading this, but the impression you gain may involve billions of neurons, many millions in the optic system alone.

We begin with the hypothesis that the Universe is fully self sufficient. All the phenomena we observe are generated by the Universe we inhabit including our own self awareness, wars, reconstructions, all the way down to the doings of electrons in computers.

All these separate worlds that we inhabit and experience are parts of the universal network which joins our private cogitations to the rest of the world.

Physics explores the structure of the world by setting up mathematical spaces and comparing to the world. The modern era began with the application of Descartes coordinate systems to space and time. This enabled us to give a space-time address to every particle in the world and then, using these addresses as a guide, following particles as they interact with one another.

[page 172]

Quantum mechanics and quantum field theory brought vast new complexity ti this old model. Art the quantum level, things do not just bang together, they communicate in a way that matches our epitome of mathematical complexity, recursive function theory (the theory of computing to us street people). Theory shows that quantum mechanical processes are naturally as complex as Turing machines, that is quantum mechanics says the Universe can compute anything computable. The Universe is a network of universal Turing machines. The hardware layer of these machines is studied by elementary particle physics, which explains the alphabet of particles which carry information around the Universe.

This sounds very complicated, but the picture becomes greatly simplified by realizing that all communication networks are governed by the mathematical theory of communication. The mathematical theory of communication, in a nutshell, says that we approach error free communication by encoding our messages into packets that are so far apart in message space that there is a negligible chance of confusing one with the other. This feature is called redundancy. We see it in action when we guess a misspelled word. Most bigger words differ in more than one letter, making them less likely to be mistaken for one another. As messages get longer they get further and further apart and less likely to be confused. It is very unlikely that there is a paragraph of text identical to this anywhere in the world unless it is copied from here.

[page 173]

At the abstract level all networks are the same so that it should not surprise us to discover features of the physical network that are repeated in human social networks.

The first thing we note in physics is that things are stable when they are in a state of stable equilibrium. Stable equilibrium means that if we push them away from their point of equilibrium they push back and resume their equilibrium point after we let them alone again.

States of stable equilibrium are generally states with the minimum of energy and the maximum of entropy. This thermodynamic observation has caused much heartache in the world because it makes it all too obvious that we are in a state of dissipation leading to the heat death of the Universe. The inevitable increase in entropy predicted by [consistent with] the second law of thermodynamics has become a model for increasing crime rates, the increasing hecticness of life, the increase in terrorism and nihilism and so on and so forth. In short, entropy has a bad press.

Entropy is saved and its soul washed clean by the mathematical theory of communication. Another word for entropy in communication theory is bandwidth, and we all know by now that it is bandwidth that makes the difference between broadband and dial-up. Just to show you my age, my first telephone connection to a remote computer operated at 300 bits per second.

Shannon conceived of information as the resolution of uncertainty. The more uncertainty a message resolves, the more information it contains. A natural measure of information is the distance between yes and no, called the bit. The game of twenty questions giving twenty yes no answers to twenty questions conveys twenty bits of information to the questioner. Ideally deployed these twenty questions should be able to discriminate between 220 = 1 million possibilities, more than enough to cover the 100 000 words in a big dictionary.

Entropy is a count of possibilities. The more possibilities, the more information is conveyed by selecting one.

So how does human society imitate the energy minimizing entropy maximizing behaviour of the rest of the world? Minimizing our energy consumption is pretty simple and obvious, but maximizing entropy is more subtle.

For a power engineer, thermodynamic entropy is a bugbear. It hides the energy in the coal that we would like to get into the power lines. Second law again

Let us imagine that the social bandwidth is adequate when everyone hears and sees pretty much everything they want to know in the public domain, when everyone can get the hearing they want and plenty of interesting stuff to want. We equate this to the equilibrium state of maximum entropy. Everyone is (at last) exactly where they want to be. This seems to be the state of affairs among physical particles. Nobody seems to be deprived? What does this tell us?

Well, communication theory tells is that the entropy of an information source is maximized when the probability of emission of each of its alphabet of symbols is equal. This is not the case with an English text such as this. The most common symbol is the space between words and after that e, . . . . One rarely sees a q. There is a lot of wasted entropy here than can be used to increase the size of the the message space and decrease the probability of error The key o stability, then is to maximize the amount of useful communication on the human network by recognizing that we are all peers [equals] on the network.

There are dissipative forces acting against our peerdom. Left to themselves the rich get richer and the poor get poorer. Not only that, but the rich tend to control social communication so that the message going round is not as pure as it could be.

Every network must have an error detecting and correcting system if it is to be stable. This is what the theory of communication is all about. Errors in general break down communication and it is the job of the technicians maintaining the network to correct such errors. The general effect of error is secrecy. While we all have a right to our privacy, we cannot trust the public network while it is subject to accidental or deliberate non-transmission of information. As peers in public, we have to be ? with one another and disclose whatever is necessary for fair trade. When all the letters in the human alphabet are equally represented in the message space, entropy, equilibrium and stability are all maximized. Life is not easy. The more minds we recruit to think out our collective destiny the better. Something like this to Eureka Street

[page 176]

The oracle machine. Washing machine stops and waits for the water to be filled, forever if necessary. All subroutines are 0-machines vis-a-vis the machine that calls them.

If the same mathematics fits two different situations we can use the one to help us understand the other and vice versa.

The opponents of democracy, like the opponents of freedom of information, claim that society will be destabilized if we let them control their own destinies and tell them the truth. Just the opposite. The truth will make us free, that is maximize entropy and stability, This is what the mathematical theory of communication says.

Ah, you will say, that is only mathematics. Reality is different. But we know a bit about reality, We know the entropy of the human brain, A trillion neurons maybe, each communicating with thousands of others at about 100 bits per second. There are nearly seven billion of these system on earth seeking sustainable peaceful and comfortable existence.

Communication theory says that the enemies of society are violent, secretive non-representative institutions that exploit people for their own benefit.

Think about transport. We all want to maximize our geographic entity by having a powerful means of personal transport always on call. I am a car driver, I seldom use public transport. There is very little where I live, and for long journeys I need a carload of stuff.

[page 177]

The advent of Wikipedia has simplified my task, so I am shutting down the local glossary and relying completely on online resources for the expansion and foundation of my story.

. . .

So we are asking ourselves what are the social implications of the spin-occupancy theorem? What does it have to do with flat-mates? Are they fermions or bosons?

All lower processes are parts of the higher (maybe the lower processes have some housekeeping functions that do not interest the users).

Users and useds. Differing in only one letter and one bit.

The used's process is isomorphic to the users process (since they are in close communication) but the meaning of the two processes may be completely different.

The notion of entropy applies to anything that is countable,. from stars to atoms and beyond in both directions, Astronomers happily talk about a gas of galaxies because it has a lot in common with a gas of molecules - a network whose dynamics are controlled by collision, ie close up communication rather than action at a distance.

Things are first discovered when they are at the limits of visibility (like an oncoming vehicle in a fog) and then we gradually learn to see them more clearly

[page 178]

LikeMme Curie, I have been boiling down tons of ore trying to distill out that radioactive flash which will light up the world. Here is the latest precipitate.

Tuesday 25 November 2008
Wednesday 26 November 2008

Say

1. Action = 0 dimension, a point like god.

2. Energy/time, 1 dimension the universal like parametrized by peroper time (unsigned)

3. photon, 2 dimensions 1 space, 1 time, bosons, which have a linear trajectory

4. fermions, 3 dimensions, 1 time, 2 space, accommodating spin and spin vectors.

5. ? 4 dimensions, 4 space structured by gravitons that couple very weakly to particles.

Then why do we stop at 4D? Because all trajectories can be accommodated withut intersection.

We guess that the heart of quantum field theory lies around processing, encoding and delay, which gives direction to causality. The simplest encoding is simply repetition, so that photons and all other quantum mechanical processes are distinguished by frequency = energy = processing rate.

The order of the layering adds properties as we go along.It may be suggestive that action has the same dimensions as angular momentum, at least once we have got a system with

[page 179]

mass, length and time. Nevertheless it seem,s nice to put action first, and consider a system of pure act that just is and does nothing. This is not observable. Observation begins when we have two, the observer and the observed, and quantum mechanics seems to be able to take care of this via exchange particles like electrons binding nuclei together.

The ordering is not very clear because I am looking from a layer much beyond the first moments of the Universe, but it seems to me that proper time goes right back to the beginning, Like a plane wave it is ubiquitous and so tells us nothing in particular. We only get interest out of a plane wave when we superpose waves with different frequencies in space (moire patterns) and/or time (musical beats and harmonies)

God, like quantum mechanics, is tota simul. Questions of before and after, delay, distance and interval do not enter quantum mechanics, But they are essential to quantum field theory. Tota simul is equivalent to parallel and quantum mechanics sees all the elements of a superposition acting together in parallel rather than successively. Succession enters from outside, as it were, from the post quantum world where there are successive 'preparations' and 'observations'.

The structure is similar to the geological layering within which we see the tree of life embedded, ie a static system (almost) the surface of which, exposed to energy, is full of the life whose remains have occasionally been fossilized.

Now we want to build spacetime as we know it from a singelt state of two fermions communicating by the exchange of a

[page180]

boson. From the special relativistic formula ds = dx - ct we assume that the spacetime interval is zero insofar as ct = dx. This situation still holds int he current Universe - the interval between the creation and annihilation of a massless particle is 0, even though it may propagate over billions of years of time and billions of light years of space.

Now what happens? This singlet is used by the next layer to build itself, How? How, in particular do we introduce spatial and temporal extension, communication delay, mass and causality. WEe go back t an early question. Why is the Universe quantized?

We see a Lorentz transformation in ordinary Minkowski space as mixing space and time, so that an inertial frame moving relative to me seems to have shorter lengths and slower clocks, to the extent that the clock on a photon appears to be stopped and the photon has no length.

The singlet state described above describes an inertial frame. We imagine it splitting into two frames, which requires the introduction of a relative velocity and a space-time interval to accomodate that velocity. IUf the velocity is less than c we also introduce mass.

At the two dimensional level the only differentiation we have is spin based, bodon or fermion, There is a relationship between spin and special relativity which, through relative velocity induces rotation between energy and momentum which may serve to differentiate particles further. Each layer of the network may be understood to introduce a new degree of freedom which, in our complex world, may

[page 181]

take many guises but which we may be able to trace back to a common root Sin, being an angular thing, might be seen to require 2 spatial dimensions. We already have angle built into quantum mechanics via complex numbers. What does this mean? We see all vectors in Hilbert space rotating at a frequency given by omega = exp(iE/h bar). This is not observable until we start superposing different energies, E. Special relativity also assumes a dichotomy between observer and observed at rest in different inertial frames.

Basically, we might see the complexity of the Universe being generated by the interactions of different observers whose difference is a result of their mutual observations, a la Trinity.

Catholicism: Control by celibate morons who have not got the slightest clue about the realities of life because they live in a finite formalism rather than a transfinite reality.

The global dialogue basically bifurcates into the sexuals and the asexuals, the former accepting the realities of death and reproduction, the latter denying them. The RCC, an antisexual, basically denies the creativity of common reality and ascribes it all to an invisible god.

The basic determinists of the direction of society are the judges, who deem some behaviour acceptable and some not.

The bandwidth of the president. How the public service funnels information into the presidential consciousness. Just like my mind funnels things into my consciousness.

[page 182]

I began my adult ife as a monk, one of those idiots who think they know everything because they don't have sex, or at least don't admit it.

Thursday 27 November 2008
Friday 28 November 2008

From a special relativistic point of view, distance = velocity ('dynamic' distance)

Saturday 29 November 2008

One might imagine that at a very early epoch action and velocity are the same thing, which then bifurcated into space-time, ie momentum-energy

Max: the creative force is talking to one another and it operates at every scale from the initial singularity to my own dialogues with other people and beyond. The creativity of the Universe lies in the non-correlation of independent entities, which are nevertheless to some degree dynamically correlated by communication, so that each leads to the growth of the other. We may take as a carefully stylized example of this the whole scientific enterprise with its readings, meetings, conferences and literature and less controlled exercises such as diplomacy and all the interactions of everyday life which have a certain material stability like houses, mealtimes and communication protocols and continuous spiritual creativity as we all respond to each other in conversation.

The power of this approach is that it allows us to use our own creative experiences of communication to

understand the creativity of the Universe that made us.

The essence of communication theory is the construction of clear and distinct symbols ('orthogonal symbols' - Zurek). Zurek

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Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Aristotle, and P H Wickstead and F M Cornford, translators, Physics books V-VIII, Harvard University Press,William Heinemann 1980 Introduction: 'Simplicius tells us that Books I - IV of the Physics were referred to as the books Concerning the Principles, while Books V - VIII were called On Movement. The earlier books have, in fact, defined the things which are subject to movement (the contents of the physical world) and analyzed certain concepts - Time, Place and so forth - which are involved in the occurrence of movement.' Book V is a further intoduction to the detailed analysis in Books VI - VIII. Book VI deals with continuity, Book VII is an introductory study for Book VIII, which brings us to the conclusion that all change and motionin the unvierse are ultimately caused by a Prime Mover which is itself unchanging and unmoved and which has neither magnitude nor parts, but is spiritual and not in space. 
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Axelrod, Robert, The Evolution of Cooperation, Basic Books 1985 Amazon.com: 'This book is a must-read not only for students (broadly defined) of the social sciences, but also for politicians and bureaucrats, especially those in charge of military and foreign affairs. Axelrod's book is a tour-de-force in multi-method approaches. Although the author is a trifle repetitive and occasionally laborious, I think the profound content of the book far outweighs the minor inadequacies of its form. At the risk of sounding like a logical positivist, I would venture to say that Axelrod's approach offers hope for a bottom-up construction of cooperation in an uncertain world without a central authority.' Reeshad Dalal 
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Axelrod, Robert, The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration, Princeton University Press 1997  
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Axelrod, Robert, and Michael D Cohen, Harnessing Complexity: Organisational Implications of a Scientific Frontier, Free Press 2000 Jacket: 'There has always been a fairly wide gap between theoretical work on complex adaptive systems and its application to real world problems confronting business and public policy. Harnessing Complexity builds a highly practical bridge bvetween these worlds, drawing on the deep insights of two thinkers who were there at the beginning of the complexity revolution. Francis Fukuyama' 
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Christie, Agatha, Murder at the Vicarage, Dodd Mead 1986 Amnazon customer review: 'Murder at the Vicarage, first published in 1930, is the book that first introduced the world to Miss Jane Marple and the cozy English village of St. Mary Mead. Every mystery fan in the world is or should be familiar with Christie's classic character of Miss Marple. This book presents her at her best and is required reading for any mystery fan. The writing is sharp, the plotting crisp and clever, there are many red herrings and the solution is very satisfying. This is Christie at her very best. Highly recommended.' Lisa Bahrami 
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Feynman, Richard, QED: The Strange Story of Light and Matter, Princeton UP 1988 Jacket: 'Quantum electrodynamics - or QED for short - is the 'strange theory' that explains how light and electrons interact. Thanks to Richard Feynmann and his colleagues, it is also one of the rare parts of physics that is known for sure, a theory that has stood the test of time. ... In this beautifully lucid set of lectures he provides a definitive introduction to QED.' 
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Lonergan, Bernard J F, Insight : A Study of Human Understanding (Collected Works of Bernard Lonergan : Volume 3), University of Toronto Press 1992 '... Bernard Lonergan's masterwork. Its aim is nothing less than insight into insight itself, an understanding of understanding' 
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Sigmund, Karl, Games of Life: Explorations in Ecology, Evolution and Behaviour, Oxford UP 1993 Jacket: 'This book takes us on a tour through the games and computer simulations that are helping us to understand the ecology, evolution and behaviour of real life - from cat and mouse to cellular automata, from the battle of the sexes to artificial life, from poker to prisoner's dilemma.' 
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Tomonaga, Sin-itiro, The Story of Spin, University of Chicago Press 1997 Jacket: 'The Story of Spin, as told by Sin-itiro Tomonaga and lovingly translated by Takeshi Oka, is a brilliant and witty account of the development of modern quantum theory, which takes electron spin as a pivotal concept. Reading these twelve lectures on the fundamental aspects of physics is a joyful experience that is rare indeed.' Laurie Brown, Northwestern University. 
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Wittgenstein, Ludwig, and Elizabeth Anscombe (translator and editor), Philosophical Investigations: The German Text with a Revised English Translation, Blackwell Publishers 2002 Amazon editorial review: 'Ludwig Wittgenstein's "Philosophical Investigations" presents his own distillation of two decades of intense work on the philosophies of mind, language and meaning. When first published in 1953, it immediately entered the centre of philosophical debate. This revised German and English edition is published on the fiftieth anniversary of Wittgenstein's death. It incorporates final revisions by G.E.M. Anscombe (1919-2001) to her original English translation.' 
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Wittgenstein, Ludwig, and David Francis Pears, Brian McGuinness, Bertrand Russell , Tractatus Logico-Philosophicus, Routledge 2001 'This as a most imortant book containing original ideas on a large range of topics, forming a coherent system, which, whether or not it be, as the author claims, in its essentials the final solution of the problems dealt with, is of extraordinary interest and deserves the attention of all philosophers.' Frank Ramsey, 'Critical Notice of L Wittgenstein's Tractatus Logico-Philosophicus', Mind, XXXII, no 128 (October 1923) pp 465-78.  
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Papers
Surowiecki, James, "The Financial Page: The Open Secret of Success", New Yorker, 84, 13, 12 May 2008, page 48. 'In the current atmosphere of economic tumult, the announcement that Toyota sold a hundred and sixty thousand more cars than General Motors in the first three months of this year might seem like a minor news item. But it may very well signal the end of one of the most remarkable runs in business history. For seventy-seven years, in good times and bad, G.M. has sold more cars annually than any other company in the world.' Surowiecki. back
Zurek, Wojciech Hubert, "Quantum origin of quantum jumps: Breaking of unitary symmetry induced by information transfer in the transition from quantum to classical", Physical Review A, 76, 5, 16 November 2007, page 052110-1--5. Abstract: 'Measurements transfer information about a system to the apparatus and then, further on, to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide a framework for 'wave-packet collapse', designating terminal points of quantum jumps and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment — the fittest quantum states that not only survive decoherence, but subvert the environment into carrying information about them — into becoming a witness.'. back
Links
Boson - Wikipedia Boson - Wikipedia, the free encyclopedia 'In particle physics, bosons are particles with an integer spin, as opposed to fermions which have half-integer spin. From a behaviour point of view, fermions are particles that obey the Fermi-Dirac statistics while bosons are particles that obey the Bose-Einstein statistics. They may be either elementary, like the photon, or composite, as mesons. All force carrier particles are bosons. They are named after Satyendra Nath Bose. In contrast to fermions, several bosons can occupy the same quantum state. Thus, bosons with the same energy can occupy the same place in space.' back
CPT Symmetry - Wikipedia CPT Symmetry - Wikipedia, the free encyclopedia 'CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity and time simultaneously.' back
Fermion - Wikipedia Fermion - Wikipedia, the free encyclopedia 'In particle physics, fermions are particles with a half-integer spin, such as protons and electrons. They obey the Fermi-Dirac statistics and are named after Enrico Fermi. In the Standard Model there are two types of elementary fermions: quarks and leptons. . . . In contrast to bosons, only one fermion can occupy a quantum state at a given time (they obey the Pauli Exclusion Principle). Thus, if more than one fermion occupies the same place in space, the properties of each fermion (e.g. its spin) must be different from the rest. Therefore fermions are usually related with matter while bosons are related with radiation, though the separation between the two is not clear in quantum physics. back
Malcolm Gladwell Annals of Innovation: Who says big ideas are rare? 'The history of science is full of ideas that several people had at the same time.' Read More back
Minkowski space - Wikipedia Minkowski space - Wikipedia, the free encyclopedia 'In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime. Minkowski space is named after the German mathematician Hermann Minkowski.' back
Oracle machine - Wikipedia Oracle machine - Wikipedia, the free encyclopedia 'In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to decide certain decision problems in a single operation. The problem can be of any complexity class. Even undecidable problems, like the halting problem, can be used.' back
Pauli exclusion principle - Wikipedia Pauli exclusion principle - Wikipedia, the free encyclopedia 'The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. It states that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement of this principle is that, for two identical fermions, the total wave function is anti-symmetric. . . . In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin. It does not follow from any spin relation in nonrelativistic quantum mechanics.' back

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