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> Note

Warped Passages

Unravelling the mysteries of the universe's hidden dimensions

Cat: SCO
Pub: 2005

Lisa Randall

  1. Preface:
  2. Entryway Passages:
  3. Restricted Passages:
  4. Exclusive Passages:
  5. Approaches to Theoretical Physics:
  6. Relativity:
  7. Quantum Mechanics:
  8. The Standard Model of Particle Physics:
  9. Experimental Interlude:
  10. Symmetry: The Essential Organizing Principle:
  11. The Origin of Elementary Particle Masses:
  12. Scaling and Grand Unification:
  13. The Hierarchy Problem:
  14. Supersymmetry:
  15. Allegro (Ma Non Troppo) Passage for Strings:
  16. Supporting Passages:
  17. Bustling Passages:
  18. Sparsely Populated Passages:
  19. Leaky Passages:
  20. Voluminous Passages:
  21. Warped Passage:
  22. The Warped Annotated "Alice":
  23. Profound Passage:
  24. A Reflective and Expansive Passage:
  25. Extra Dimensions:
  26. Conclusion:
  1. 序文:
  2. 入り口の通路:
  3. 限定的な通路:
  4. 排他的な通路:
  5. 理論物理学へのアプローチ:
  6. 相対論:
  7. 量子力学:
  8. 素粒子物理学の標準モデル:
  9. 実験的な幕間:
  10. 対称性:
  11. 素粒子質量の起源:
  12. スケーリングと大統一:
  13. 階層問題:
  14. 超対称性:
  15. 急速な紐の通路:
  16. 脇道の通路:
  17. 賑やかな通路:
  18. 過疎の通路:
  19. 漏れる通路:
  20. たっぷりした通路:
  21. ワープした通路:
  22. ワープした注釈付きのアリス:
  23. 深淵な通路:
  24. 収縮・膨張する通路:
  25. 余剰次元:
  26. 結論:


; ADD model; Anarchic principle; Conceptual leap; Effective theory; Elecronvolt; General Relativity; Grand Unified Theory; ; Hierarchy problem; Higgs; Hořava-Witten Theory; Internal symmetry; Kaluza-Klein; M-theory; Multiverses; Planck length; Photoelectric effect; Photon; Perturbation theory; Quantum Chromo Dynamics; Rolled-up dimensions; Standard Model; String Theory; Superstring Theory; Theory of Everything; Virtual Particles; Wavefunction;
  • This book contains various interesting episodes behind the development of modern physics, not only explaining the essence of particle physics and extra dimensions.
  • It is diffcult for 2D Flantlander to imagine 3D cube. Similarly we, 3D animal, have no concrete terms to understand extra dimensions, other than mentioning vague expressions such as; extra, additional, invisible, unseen, bulk, mass, rolled-up, curled-up, hidden, small, compact, subtlety, counterintuitive, mysterious, ...
Original resume


0. Preface:

  • The tone (of books about science) often seemed condescending to readers, overly worshipful of scientists, or boring. I felt the authors mystified the results or glorified the men who found them, rather than describing science itself and the process by which scientists made their connections.
  • I've bulleted the points that I'll refer to later on when I present more recent ideas about extra dimensions.
    • The recent work on extra dimensions relies on both more traditional and more modern theoretical physics concepts to motivate the questions it answers and its methods.
    • I've described some newer, or speculative notions that have been studied for the last thirty years - namely supersymmetry and string theory.
    • Supersymmetry is an extension of known particle physics; String theory is different, which is based on theoretical questions and ideas.
    • I'm an agnostic on this subject. But string theory has been a rich resource for new ideas.
    • Extra dimensions can be infinite in size yet remain unseen, or that we can be living in a three spatial dimensional sinkhole in a higher dimensional universe.
  • An example of quasirystalline pattern (> Right figure):
    • Elegant way of this pattern; 3D shadow of a higher dimensional crystalline pattern. What looked like a completely inexplicable pattern in 3D reflects an ordered structure in the higher dimensional world.
  • Physicists today speculate that theories of extra dimensions also will illuminate connections in particle physics and cosmology.
    • String theory, which postulates that the most basic units in nature are not particles but fundamental, oscillating strings; which requires more dimensions of space.
    • Role of branes, membrane-like objects within string theory, which are as essential to the theory as strings themselves.
  • 3D pocket of space:
    • Not only are we not in the center of the universe, as Copernicus shocked the world 500 years ago, but we just might be living in an isolated neighborhood with three spatial dimension that is a part of a higher-dimensional cosmos.

0. 序文:

  • 今までの科学書は権威的で退屈
  • ここでは最新の超次元についての核心にずばり迫る。
    • 超次元については、物理学の伝統を最新の理論に依存している。
    • ここではより新しい概念を提起する。
    • 超対象は既存の素粒子論の延長にあるが、弦理論は理論に基づいている。
    • それについては不可知論の立場ではあるが、弦理論は魅力的なアイデアを提供してくれる。
    • 超次元の大きさは目に見えない無限大かも知れないし、我々は高次元宇宙の中の三次元の穴の中に住んでいるのかも知れない。
  • Penrose tiling:
  • PenroseTiling

>Top 1. Entryway Passage:

  • Three from Two:
    • We can use 2D information to construct 3D. We often use to reduce higher dimensional objects to lower dimensions - slicing, projection, holography, and sometimes just ignoring the dimension.
    • A single eye constructs a 2D projection of 3D reality. You need two eyes to reproduce 3D.
  • A hypercube:
    • consists of 8 cubes attached in 4D space. (>a few projections)
  • Practical projection (Medicine, etc.):
    • X-ray; records 2D project.
    • CAT (Computer assisted tomography) scan combine multiple X-ray imaged to reconstruct 3D representation.
    • MRI (Magnetic resonance imaging) scan, on the other hand, reconstructs 3D object from slices.
    • Holograph: records relations between light in different places.
    • Stereo; let you hear where instruments were being played in relation to each other.
    • Ink on a paper is 3D, but we lose nothing by thinking of it as 2D.
    • Wire; looks 1D, but is has 2D cross section and therefore 3D in all.
  • Conceptual leap:
    Nothing wrong with ignoring extra dimension: selecting relevant information and suppressing details is the sort of pragmatic fudging; we can focus on the issue of interest.
    • Newton's law: didn't need the details of general relativity.
    • Studying a cell; don't need to know about quarks inside the proton.
  • >Top Effective theory:
    • There is nothing wrong with ignoring an extra dimension that's too small to be seen.
      • Selecting relevant information and suppressing details; so that you can focus on the issue of interest, and not obscure it with inessential details. (conceptual leap)
      • Concentrate on the particles and forces that have "effects" at the distance in question.
    • >Top we formulate observations in terms of the things that are actually relevant to the scales we might detect. (Conceptual leap)
      • This practie is not scientific fraud, but a way of disregarding the clutter of superfluous information.
      • If something is beyond the resolution of the scales, you don't need its detailed structure; it is an effective way to obtain accurate answers efficiently.

1. 入り口の通路:

  • 3Dの表現方法:
    • スライス、投影、ホログラフ、追加次元の無視
    • 単眼は2D像。3Dにするには両眼が必要
    • インク、ワイアも本来は3D
  • 超立方体; 超方形 chāo fāngxíng
  • hypercube3
  • 概念上のジャンプ:
    • ニュートン力学
    • 細胞
  • 有効理論:
    • 必要な情報に注目するのは非科学的ではない。

>Top 2. Restricted Passage: Rolled-up Extra Dimensions

  • Rolled-up dimensions in physics:
    • String theory, the most promising candidate for a theory combining quantum mechanics and gravity.
  • Einstein's theory:
    • Early 20C, Einstein's theory of relativity opened the door of extra dimensions of space; it works equally well for three or four or ten dimensions. Why, then, do there seem to be only three?
    • In 1919, Polish mathematician Teodor Kaluza recognized this possibility and boldly proposed a fourth spatial dimension, a new unseen dimension of space.
      • Kaluza's goal with this extra dimension was to unify the forces of gravity and electromagnetism.
      • Kaluza wrote his paper in 1919. Einstein wavered about the merits of the idea, delayed the publication of Kaluza's paper for two years, but eventually acknowledged its originality. Yet Einstein still wanted to know what this dimension was. Where was it and why was it different? How far did it extend?
      • Oscar Kleing proposed that the extra dimension would be curled up in the form of circle, just $10^{-35}$ m
    • >Top Planck length; $1.616 × 10^{-35}$ m
    • Daily life of objects:
      • Paint of a wall; paint's depth
      • Clothesline's thickness
      • A hose over a football field
  • >Top Rolled-up dimensions:
    • Focus on any point along the infinite dimension. Notice that at each and every point there sits the entire compact space, namely circle. The hose consists of all these circles glued together.
      • There are two infinite dimensions rather than one, plus a single additional dimension curled up into a circle.
      • When tow out of four dimensions are curled into a donut, you have a donut at every point in space.
      • Compact spaces: Kalabi-Yau manifolds
      • Planck-length extra dimensions would have no visible trace of their existence.
  • Dimensions have to be small to be invisible?
    • Could an extra dimension possibly extend for ever without our seeing it?
      • Even the radical possibility of an infinite extra dimension cannot be excluded if it is sufficiently different from the three familiar infinite dimensions.
      • >Top 4D Kaluza-Klein universe with a single tiny, rolled-up dimension would appear to us to have one dimension fewer than the four it actually has.
      • In an effective theory all that matters are the things that you can actually perceive.
  • Gravitational force:
    • Radial lines extending outwards from the planet's center:
      • the density of these lines determines the strength of gravitational attraction that the planet

2. 限定的な通路: 巻き上げられた余剰次元

Order of magnitude Range Unit Example
Planck length $10^{-35}$ ℓp Quantum foam
Sub atomic $10^{-15}$ atto-m electron, quark
Atomic-cellular $10^{-12}$ femto-m proton, neutron
$10^{-9}$ pm H atom
$10^{-6}$ nm DNA helix, virus
Human scale $10^{-3}$ μm bacteria
$10^{0}$ mm mosquito
$10^3$ m human
$10^6$ km Mt. Everest
Astro-nomical $10^9$ Mm Earth
$10^{12}$ Gm Sun
$10^{15}$ Tm solar system
$10^{18}$ Pm ly
$10^{21}$ Exa-m galactic arm
$10^{24}$ Zetta-m Milky Way
  • 巻き上げられた次元
  • rolledup_dimension
  • Gravitational force:

>Top 3. Exclusive Passage: Branes, Braneworlds, and the Bulk

  • Slices or brane:
    • 3D world could be a 3D slice of a higher dimensional world.
      • The word "membrane" motivated the choice of the word "brane" because membranes, like branes, are layers that either surround or run through a substance.
      • Some branes are slices inside the space, but others are slices that bound space, like slices of bread in a sandwich.
      • Membrane has 2D, but branes can have any number of dimensions.
  • A pipe with a square cross-section:
    • An infinitely long pipe of this type would have four infinitely long straight walls; two of which are bounded on either side by walls and one that extends infinitely.
    • Reflective boundary condition: if things bounce back from a brane, energy is not lost; nothing goes beyond the branes.
    • When things get to a boundary brane, they bounce back, just as billiard balls. (reflective boundary condition) Nothing goes beyond the branes.
    • Such branes would have lower dimension than the full space.
  • Trapped on branes:
    • inside a black hole
    • drops of water on a 2D shower curtain
    • bacteria trapped between microscope slides
    • Sam Loyd's fifteen game
    • Brane-bound objects never venture into the extra dimensions that extend off the brane.
  • Braneworlds:
    • Gravity would extend into the extra dimensions, but stars, planets, people, and everything else that we sense could be confined to a 3D brane. A brane might be our habitat.
    • Our senses are attuned to the chemistry, light, and sound surrounding us. Because fundamental forces and particles are likely to be different, the creatures of other branes, should they exist, are unlikely to bear much resemblance to the life of our brane.
    • >Top The universe can contain multiple branes that interact only via gravity or don't interact at all. Such set-ups are sometimes called multiverses.
      • Brane-bound particles would never have direct contact with particles bound on another brane.
    • Dark matter and dark energy; we surmise from their gravitational effects but whose identity is a mystery.

3. 排他的な通路: ブレイン、ブレイン世界、バルク:

  • スライスまたはブレイン:
    • ブレインとは空間の中のスライス、または空間を区切るスライス
    • 膜は2Dだが、ブレインは何次元にもなり得る。
  • Lower-demensional surface:
  • lowerdimensionsurface
  • ブレインに囚われて
    • ブラックホール内
    • 2Dのシャワーカーテンについた水滴
    • 薄片に挟まれた細菌
    • ブレインに囚われた物体は敢えて超次元方向へ動こうとしない。
  • ブレイン世界:
    • 重量は超過次元まで作用するが、我々が感知する物体は3Dブレインに閉じていう。
    • 多元的宇宙
    • ダークマター、ダークエネルギー

>Top 4. Approaches to Theoretical Physics:

  • Ultimate goal; to find a simple, elegant, unifying theory capable of explaining all observations.
  • Top-down approach:
    • Theorists start with the theory to be correct and try to derive its consequences so that they can connect it to the much more disordered world we observe.
    • Platonic approach; tries to gain insights from more fundamental truth.
    • Young Einstein; rooted his work in experiments and physical reality; so-called thought experiments were grounded in physical situations.
    • Karl Schwartzschild first derived black holes as a mathematical consequence of general relativity.
  • Bottom-up approach:
    • Model builders try to deduce an underlying theory by making connections among observed elementary particles and their interactions. They search for clues in physical phenomena.
    • Aristotelian approach; rooted in empirical observations.
    • Old Einstein; changed his approach after learning the value of mathematics when developed general relativity.
  • >Top String Theory:
    • basic premise is that strings, not particles, are the most fundamental objects of nature. The particles we observe are mere consequences of strings; arise from the different vibrational modes of an oscillating string.
    • is defined at an energy scale that is about ten million billion times larger than those we an experimentally explore with our current instrument.
    • its invisible extra dimensions have to be different from the 3D that we see.
    • string theory's gravity operates in six or seven additional dimensions of space.
    • Why aren't they all the same? Discovering how and why nature hides string theory's extra dimensions would be a stunning achievement.
  • We are all searching for the language of the universe:
    • Theorists focus on the inner logic of the grammar; might aspire to grasp the subtleties of Italian literature, but run the risk of starving to death before learning how to ask for dinner!
    • Model builders focus on the nouns and phrases that are most useful.; model builders would know how to ask for lodging and acquire the vocabulary that would be essential to finding they way around.
    • Now; we can certainly ask where the extra dimensions are, and why we haven't seen them.; whether these unseen dimensions could have any import in our world.
  • The Heart of Matter:
    • Physical objects around us appear to be continuous and uniform, but in reality they are not.
    • Particle physics; smaller distance reveal truly novel phenomena.; new worlds operate via more fundamental physical laws.
      • Nucleons; protons and neutrons.
        • Proton: made of 2 up quarks and 1 down quark
        • Neutron: made of 1 up quark and 2 down quarks
        • bound together through the strong force.
      • Electron: cannot be divided into smaller particles and contains no substructure within.
  • >Top Standard Model:
    • Stephen Weinberg coined the term Standard Model.; fundamental building blocks of matter.
    • up quark, down quark, and electron lie at the core of matter.
    • we now know that there exist additional, heavier quarks and heavier electron-like particles that are nowhere to be found in ordinary material.
      • muon, with precisely the same charge as the electron, ha a mass that is 200 times greater than electron's.
      • tau, which also has the same charge, has a mass that is 10 times greater still.
      • heavier elementary particles r not constituents of matter; were a part of the early universe immediately after the Big Bang.
    • No one know why the heavy Standard Model particles exist; why their masses are so different from those of the constituents of more familiar matter.
      • Could there be others yet detected?
      • Why is gravity so much weaker than the other known forces.
      • We're confident that more fundamental structure awaits discovery and that the search for more fundamental principles will be rewarded.
  • Braneworld ideas:
    • have provided new insights into general relativity, particle physics, and string theory.
    • we don't yet know which of these ideas correctly describe nature.; but we'll ultimately discover that branes are a part of he cosmos, and we are confined to them.

4. 理論物理学へのアプローチ:

  • 究極のゴール
  • トップダウンアプローチ
    • 理論家
    • プラトン的
    • 若きアインシュタイン
    • シュヴァルツシルト
  • ボトムアップアプローチ
    • モデル構築者
    • アリストテレス的
    • 古きアインシュタイン



  • ひも理論


  • 宇宙の原語
    • 理論家: 文法の内部構造
    • モデル構築者: 実用的な単語重視


  • Standard Model of Physics (標準モデル): >下図


  • Field os phtysics how they are connected:



1st generation 2nd generation

3rd generation

Force-carrying gauge bosons  

u-qk (2.3MeV)

c-qk (1.3 GeV) t-qk (173 GeV) gluon (0) Higgs (126 GeV)

d-qk (4.8 MeV)

s-qk (95 MeV) b-qk (4.3 GeV) $\gamma$ (0)  
electron (0.5 MeV) muon $\mu$ (106 MeV) tau $\tau$ (1.8 GeV) Z-bozon (91 GeV)  
$\nu_e$ (<2.2 eV) $\nu_{\mu}$ (<0.2 MeV) $\nu_{\tau}$ (<16 MeV) W-boson (80 GeV)  

>Top 5. Relativity; The Evolution of Einstein's Gravity:

  • Beginning of 19C: Lord Kelvin said, "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement."; which was famously incorrect.
    • very soon, relativity and quantum mechanics revolutionized physics.
  • Special Relativity:
    • Einstein's first insight about reference frames and relativity came from thinking about electromagnetism.
    • Einstein realized that if there were an aether, there would also be a preferred observational vantage,
    • Einstein's critical insight was that ideas about time had to be reformulated; space and time could no longer be considered independently.
    • The laws of special relativity that follow lead to all the surprising consequences, such as time dilation, the observer dependence of simultaneity, and Lorentz contraction of a moving object.
  • Equivalence principle:
    • the effects of acceleration cannot be distinguished from those of gravity.
    • follows from the equivalence of inertial and gravitation mass, two quantities that in principle could have been different from each other.
    • Ireland's Cliffs of Moher, "It's not the fall that kills you, but the crash when you stop."
  • Tests of General Relativity:
    • Gravitational redshift of light:
      • The rising ball slows down as it moves against the force of gravity. But the ball's energy is not lost; it is converted into potential energy, releasing kinetic energy, or energy of motion.
      • a photon loses momentum as it escapes from a gravitational field; the photon loses kinetic energy but gains potential energy as it fights its way out of the gravitational field.
    • Bending of light:
      • the famous relation $E=mc^2$ means that energy and mass are closely connected.
      • the sun's gravity influence mass, and likewise affects the trajectory of light; first confirmed during the solar eclipse of 1919.
      • It is one of the tools that was used to probe the distribution of mass in the universe and look for dark matter in the form of small, burn-out stars.
    • Gravitational lensing;
      • light emitted by a bright star will bend when it passes by the dark star. Light passing on the left will bend in the opposite direction that light passing on the right and light passing on the top will bend in the opposite direction than light passing on the bottom; this will create multiple images of a bright object behind a dark star.
    • Graceful curves of the universe:
      • Einstein no longer saw gravity as a force that acts directly on an object. Instead, he described it as a distortion of the geometry of spacetime that reflects the different accelerations required to cancel gravity in different places.
      • the force of gravity is understood in terms of the curvature of spacetime, which in turn is determined by the matter and energy that are present.
  • Curved space and curved spacetime:
  • >Top Theory of General Relativity:
    • Free fall along a geodesic:
      • in curved spacetime, the geodesics of different observers will in general be different.
      • A massive object distorts the surrounding space, creating a gravitational field.
      • In reality, the full 4D spacetime are warped. Time is warped.
    • $E=mc^2$ (Einstein's equations)
      • General relativity has further advantage that it incorporales any type of energy - including that of the gravitational field itself - into the distribution of matter and energy.
      • Distorting space: >Fig.
        • the space surrounding the ball looks like 2D. But really, the full 4D spacetime are warped. Time is warped because it too is a dimension from the vantage point of special & general relativity.
        • it' important to keep in mind is that the object distorting spacetime can have any number of dimensions.; a brane will play the role of the sphere in this picture.
    • Einstein's equations changed the outlook for cosmologists. Now, if scientists new the matter and energy content of the universe, they could calculate its evolution.
      • In an empty universe, space would be completely flat, with no ripples or undulations - no curvature at all.
      • But when energy and matter fill the universe, they distort spacetime, producing interesting possibilities for the universe's structure and behavior over time.
    • General relativity has lots of consequences:
      • eliminated the annoying action-at-a-distance of Newtonian gravity, which asserted that an object's gravitational effects would be felt everywhere as soon as it appeared or moved.
        • With general relativity, we know that before gravity can act, spacetime has to deform. This process does not happen instantaneously, IT takes time. Gravity waves travel at the sped of light. Gravitational effects can kick in at a given position only after the time it takes for a signal to travel there and distort spacetime.
    • Black holes; the geometry of spacetime is extremely distorted;
      • Karl Schwarzschild discovered that black holes were a consequence of Einstein's equations almost immediately after general relativity's development.
      • It was not until 1960s that they could be real things in our universe.
    • GPS system:
      • to calculate position within a meter, we must measure tie to better than one part in $10^{13}$
      • But even if we had perfect clocks, time dilation wold slow them down by a bout one part in $10^{10}$; this error would be 1000 times too big for our desired GPS system.

5. 相対論: アインシュタインの重力の進化

  • 重力赤方偏移




  • 一般相対性理論:
  • Distorting space:


>Top 6. Quantum Mechanics: Principled uncertainty, Principal uncertainties, and the uncertainty principle:

  • Quantum mechanics, counterintuitive as it is, fundamentally altered the way scientists view the world.
    • Much of modern science evolved from quantum mechanics; statistical mechanics, particle physics, chemistry, cosmology, molecular biology, evolutionary biology, and geology were all either invented or revised as a result of its development.
      • Modern world; such as computers, DVD players, and digital cameras wouldn't be possible without the transistor and modern electronics relied on quantum phenomena.
      • Physicists developed quantum theory to interpret experimental results that classical physics could not explain. And quantum theory, in turn, suggested further experiments with which to test hypotheses.
    • Shock and awe:
      • Scientists had to suspend their disbelief before they could accept the quantum mechanical premises.
      • Even theoretical pioneers, such as Max Planck, Erwin Schrödinger, and Albert Einstein, never really converted to the quantum mechanical way of thinking.
        • Einstein voiced, "God does not play dice with the universe."
      • Radical nature of the scientific advances in early 20C reverberated in modern culture.
      • Quantum mechanics is difficult to understand because its consequences are so counterintuitive and surprising. Its fundamental principles run counter to the premises underlying all previously known physics - and counter to our own experiences.
      • Quantum effects generally become significant at distance of about an angstrom, the size of an atom.
        • We see only huge aggregates of atoms, so many that classical physics overwhelms quantum effects.
        • Werner Heisenberg explained, "We simply have to remember that our usual language does not work any more, that we are in the realm of physics where our words don't mean much."
  • Old quantum theory:
    • began in 1900; Max Planck suggested that light could be delivered only in quantized units.
      • the fundamental unit is equal to a quantity, now known as Planck's constant, $h$, multiplied by the frequency, $f$.; the energy of light could be $if, 2hf, 3hf,$ and so on.
    • Blackbody ultraviolet catastrophe:
      • Classical calculations predicted that far greater energy would be emitted in high-frequency radiation.
      • Measurements showed that different frequencies do not contribute democratically to blackbody radiation; the very high frequencies contribute less than the lower ones. Only the lower frequencies emit significant energy. This is why radiation objects are "red-hot" and not "blue-hot."
      • Only frequencies below some specific upper limit could contribute to radiation from a blackbody.
  • >Top Photoelectric effect:
    • Below the threshold, no electrons are emitted from the metal regardless of the light intensity or the length of time of exposure.
    • $E=\franc{ha}{\lambda}$
      where, $E$: work function, $\lambda$: cut-off wavelength
    • 1921: Einstein was awarded Nobel Prize in Physics.
    • But he was reluctant to accept that these quanta were actually massless particles, which carried energy and momentum but had no mass.
    • 1923: Compton scattering:
      • a quantum of light hits an electron and is deflected.
      • Measurement showed that the quanta of light behaved precisely as if the quanta were massless particles that quanta were indeed particles (now called photon)
  • Quantization and the atom:
    • Atom's true properties were not accepted until the beginning of 20C.
    • 1897, J.J. Thomson identified electrons and proposed that the electron was an ingredient of the atom; plum pudding model, positively charged component spread the atom with negatively charged electrons (the pieces of fruit) embedded inside.
    • 1910, Ernest Rutherford; discovered a hard, compact atomic nucleus much smaller than the atom; shooting alpha particles from Rn-222 at atoms.
      • "It was quite the most incredible event that has ever happed in my life, as if you fired a 15 inch shell at a piece of tissue paper and it came back and hit you."
      • the positive charge was not spread throughout the atom, but was confined to a much smaller inner core (nucleus)
      • James Chadwick; the discover of the neutron.
    • Niels Bohr; electrons acting as if they were waves, oscillated up and down sa they circulated about the nucleus.
      • It can only have a wavelength that would permit the wave to go up and down a fixed number of times.
      • He made his assumption simply to account for the stable electron orbits.
  • Electron Quantization:
    • an electron's orbit had to have a radius that fit a formula he proposed.
      • he made his assumption simply to account for the stable electron orbits.
      • they oscillated up and down once over a fixed distance;
      • that distance is the wavelength; that would permit the wave to go up and down a fixed number of times.
    • Bohr could explain why photons were emitted or absorbed only a the measured frequencies.
  • >Top Max Born:
    • proposed a surprising interpretation; that the wave was a function of position whose square gives the probability for finding a particle. (wavefunction)
    • We cannot know the particle's exact location; only specify the probability of finding it somewhere.; probability wave
  • Schrödinger:
    • developed the wave equation that shows the evolution of the wave associated with a quantum mechanical particle.
    • if you were to perform the double-slit experiment with electrons, you would see what Young saw for light: a wave lie pattern on the screen behind the slits
  • The uncertainty principle and special relativity:
    • The position-momentum uncertainty principle says the product of uncertainties in position and in momentum must exceed Planck's constant.
    • According to special relativity, when momenta are high, so are energies.
    • Combining these two facts tells us that the only way to explore short distances is with high energies.
  • >Top Elecronvolt (eV):
    • an electronvolt is the energy required to move an electron against a potential difference, such as could be provided by a very weak batter of 1 volt.
      • Particle physics often use the units (MeV, GeV, TeV) to measure not just energy, but mass, momentum.
      • The proton mass is 1 GeV.
  • Weak scale energy; 250 GeV = $10^{-16}$ grams.
    • Weak scale length is $10^{-16}$ cm; the maximum distance over which particles can influence each other through this force
  • Planck scale energy: $(M_{Pl}): 10^{19}$ GeV
    • Planck scale length: $10^{-33}$ cm; extremely small distance.
    • The Planck scale energy determines the strength of gravity; it is the energy that particles would have to have for gravity to be a strong force.
  • Bosons and Fermions:
    • those particles could be fundamental particles such as the electron and quarks.
    • whether an object is a boson or a fermion depends on a property called intrinsic spin.
      • Photon is a boson and has a spin -1.
      • Quantum spin can take the value 0, 1, 2 or any integer number units
    • Fermions have half-integer values such as $\frac{1}{2}, \frac{3}{2}$
      • Protons, neutrons, and electrons are all fermions with spin $-\frac{1}{2}$
      • The Pauli exclusion principle states that two fermions of the same type will never be found in the same place; so matter can't just collapse.
    • Bosons act in exactly the opposite fashion to fermions.; they can be found in the same place.
      • they are like crocodiles - they prefer to pile up on top of one another.
      • Two light beams can shine in exactly the same place. In fact, lasers are based on this fact.; produce their strong, coherent beams.
    • 4e condensate (BEC):
      • a state of matter of a dilute gas of bosons cooled to close to 0 K (-273.15ºC)
      • 1938: superfluidity and superconductivity.

6. 量子力学: 不確定問題

  • Backbody spectrum:
    x: frequency
    • lower waverlength corresponds to lower frequncy, or lower energy, or lower temperature.)
      y: relative intensity.
    • Universe now: 2.7 degrees above aboslute zeor.


  • Blackbody: emits most of its radiation at low frequnecies, and high frequencies be cut out.
    • Radiation pattern (Spectrum):
    • An analogy: ordering dessert
  • Photoelectric effect (光電効果):







  • Standard Model:


>Top 7. The Standard Model of Particle Physics: Matter's most basic known structure:

  • Quarks; the building blocks of the proton:
    • 1 $cm^3$ pea in a 2m × 1m × $\frac{1}{2}$m mattress takes up one-millionth; a quark occupies in a proton.
    • the Standard Model and its key ideas will contribute to a deeper understanding of matter's fundamental structure and the way physicists think about the world today.
  • 1819 Hans Oersted;
    • deduced that there should be a single theory describing both electricity and magnetism; must be two sides of the same coin. (electromagnetism)
    • 'Field' is any quantity that permeates space; the value of the field at any location tells us how intense the field is there.
    • first half 19C, Michael Farady; introduced the concepts of electric an magnetic fields; ubiquity of electric fields; action at a distance.
      • it would be extraordinary if something in one place could instantly affect another object some distance away. How would the effect be communicated?
      • The electric and magnetic fields change no faster than the finite speed of light allows.
      • Maxwell's electromagnetic theory gave Einstein the insight about the constant speed of light that instigated his monumental work.
    • The quantum electromagnetic theory attributes the electromagnetic force to the exchange of photon
  • >Top Photon:
    • the first example of a gauge boson, elementary particle that is responsible for communicating a particular force.
    • Two electrons enter the interaction region, exchange a photon, and then emerge with their resultant paths. (the Feynman diagram) >Fig.
      • Richard Feynman; QED (Quantum ElectroDynamics): predicts how photon exchange produced the electromagnetic force.
    • Quantum field theory: particles can be produced or destroyed anywhere and at any time.
  • In 1960s: Sheldon Glashow, Steven Weinberg, Abdus Salam:
    • independently developed the electroweak theory; the of the weak force.
    • Weak gauge bosons produces the effects of the weak force, just as photon exchange communicates electromagnetism; W+, W, Z
  • The weak force; distinguishes left from right, or violates parity symmetry; chirality
    • When a neutron interacts with a weak gauge boson, a proton might emerge.
      • a down quark contained in the neutron into an up quark contained in the proton.
    • Violation of parity symmetry is not the only novel property of the weak force.
      • over a very short distance ($10^{-16}$ cm); quite unlike gravity and electromagnetism.
      • The photon coveys the electromagnetic force to large distances.
    • the weak force can actually convert one particle type into another.
      • A neutron interacts with a weak gauge bozon, a proton might emerge. <Fig>
      • However, because neutron and proton have different masses and carry different charges, the neutron must decay into a proton plus other particles, so as to conserve charge, energy, and momentum.;
    • Beta decay:
      • it produces but also an electron and a neutrino (antineutrino) <Fig>
        • neutrino has no electric charge and does not decay, it was invisible to detectors and no one knew it existed.
        • without the neutrino, beta decay looked as if it wouldn't conserve energy.
        • Enrico Fermi; named neutrino.
  • >Top Quarks and the Strong Force:
    • Quantum Chromo Dynamics (QCD); the last of the Standard Model force that we can explain with gauge boson exchange.
    • 1950s-60s: many particles were discovered successively; collectively, these particles are called hadrons.
      • Hadrons were all much more massive than the electron; comparable in mass to the proton (2,000 times the electron's mass)
      • Light quarks and leptons found in matter; heavier quarks and leptons also exit.
      • These heavy particles are unstable; they decay to lighter quarks and leptons.
      • There are three generations, each of which contains successively heavier versions of each particle type.
        • these particle varieties are known as flavors.
    • Gell-Mann proposed three varieties of quark; up, down, and strange.
      • the strong force was the last force to be understood.
      • Proton=2 up quarks ($+\frac{2}{3}$)+1 down quark ($-\frac{1}{3}$)=+1 electric charge.
      • Neutron= 1 up quark + 2 down quark = zero electric charge.
      • Quarks never appear as free unaccompanied objects.
  • The known fundamental particles:
    • three of the four known forces: electromagnetism, the weak force, and the strong force.
    • the Standard Model; identified by their charges and by handedness.
    • these particles as either quarks or leptons (E.g.: electrons and neutrinos)
    • all of the lighter stable quarks and leptons have heavier replicas.
    • Muon, first seen in cosmic rays, was nothing other than a heavier version of electron (200 time heavier); is negatively charged.
      • Muon is unstable and quickly converts into an electron and two neutrinos.
    • Muon decay:
      • muon turns into a muon neutrino and a virtual $W^-$ gauge boson, which then converts to an electron and ana electron antineutrino.
  • The 2nd generation particles have charges identical to those oft heir first generation counterparts but are heavier.

7. 素粒子物理学の標準モデル:

  • Feynman Diagram


  • Interaction with a $W^-$ gauge bozon


  • Beta decay:


  • Muon decay:


>Top Quarks: experience strong force Leptons
1st generation
3 MeV
7 MeV
3 MeV
7 MeV
$e$ neutrino-L
0.5 MeV
0.5 MeV
2nd generation Charm-L
1.2 GeV
120 MeV
Charm-R 1.2 GeV Strange-R
120 MeV
$\mu$ neutrino-L
106 MeV
106 MeV
3rd gneration Top-L
174 GeV
4.3 GeV
174 GeV
4.3 GeV
$\tau$ neutrino-L
1.8 GeV
1.8 GeV
  Quarks-L: weak force   Leptons-L: weak force  
  • Three generations of the Standard Model; Left- and Right- handed quarks and leptons are listed separately.
  • Each column contain particles with the same charge (different flavors)
  • The weak force can change elements of the first coulmn into elements of the second, and the fifth comun into the sixth.
  • The quarks experience the strong force, whereas the leptons do not.

>Top 8. Experimental Interlude; Verifying the Standard Model:

  • High-energy physics; high-energy particle collider
    • Experiments at high-energy colliders will tell us about fundamental laws of nature that cannot be observed in any other way.
  • Top Quark Discovery:
    • the Standard Model would be inconsistent without it; as recently as 1995, no one had ever detected one.
    • Bottom quark, which weights at 5 times the mass of a proton, was discovered in 1977.
    • Top quark was remarkably heavy compared with the other quarks.
    • Finally appeared after 20 years of searching; there were trillion collision events that didn't contain a top quark.
  • Tevatron (<TeV) located at Fermilab, in Batavia, Illinois, 50 km west of Chicago;
    • Many heavy particles are unstable and decay almost immediately.
    • Experiments search for visible evidence of a particle's decay products, rather than the particle itself.
  • CERN (Conseil Européen pur la Recherche Nucléaire) in Switzerland:
    • Tim Berners-Lee; came up with WWW and HTML.
    • LHC (Large Hadron Collider) will be built within a decade; will be able to reach 7 times the present energy of the Tevatron.
    • To determine the precise value of Z bozon's mass; they had to take into account everything that might affect the value of its energy.
      1. Tides in Lake Geneva; which sightly altered the distance over which the electrons and positrons traveled inside the collider
      2. TGV, the express train that ravels between Geneva and Paris; power spikes associated with the French DC current that slightly disrupted the acceleration; there was inevitably a strike, treating to a spike-free day!

8. 実験的な幕間:

  • the Standard Model:
  • Fermions half integer spin
      Qaurks U, D, C, S, T, B (quark/antiquark)
      Leptons Electron, Positron,
        Muon, Antimuon
        Tau, Antitau
        Electron neutrino, Electron antineutrino
        Muon neutrino, Muon antineutrino
        Tau neutrino,
    Tau antineutrino
    Bosons integer spin
      Gauge Photon, Gluon, W & Z bosons
      Scalar Higgs boson


>Top 9. Symmetry:

  • the Standard Model:
    • is a theory of quarks, leptons, and weak gauge bosons (Ws and Z); all have mass.
    • if matter had been truly massless, it wouldn't form nice, solid objects, and structure and life in the universe.
  • Symmetry:
    • is a sacred word to most physicists.
    • Christian cross, Jewish menorah, Dharma wheel of Buddhism, Crescent of Islam, Hindu mandala.
    • Symmetry can be found in most paintings sculpture, architecture, music, dance, and poetry; Islamic art, Taj Mahal
    • Symmetry is a natural player; when a physical system has symmetry, you can describe the system on the basis of fewer observations than if the system had no symmetry.
      • if a system possesses rotational and translational symmetries, physical laws apply the same way in all directions and in all places.
      • Many physical theories, such as Maxwell's laws of electrodynamics, and Einstein's theory of relativity, are deeply rooted in symmetry.
  • >Top Internal symmetry:
    • tells us the physical laws act the same way on distinct, but effectively indistinguishable, objects.
    • Particle physics; treat particles and the fields that create them as interchangeable. (flavor symmetry)
      • Sometimes exploiting even slightly imperfect symmetries helps us to compute sufficiently accurate results.
  • Gauge bosons, Particles, and Symmetry:
    • Each of the internal symmetries can be preserved only if it transforms both the gauge bosons and the particles with which they interact.
      • Rotations wouldn't be symmetry transformations if the acted on some objects but not others. If you rotate the top wafer of an Oreo cookie, but not the rest of it, you would pull it apart. The Oreo cookie would look the same after a rotation only if you weer to rotate the entire thing simultaneously.
    • The force that will interest us most is the weak force. the internal symmetry associated with the weak force treats the three weak gauge bosons as equivalent.
      • It also treats particles pairs such as electron and neutrino, or up and down quarks, as equivalent.
  • Symmetries are important to the theory of forces because the simplest workable theory of forces includes a symmetry associated with each force.
    • Those symmetries eliminate unwanted particles. They also eliminate the false predictions that the simplest theory of forces would otherwise make about high-energy particles.

9. 対称性:

  • Oreo Cookie:

>Top10. The origin of elementary particle masses: Spontaneous symmetry breaking and the Higgs mechanism

  • Symmetries are important, but slightly imperfect symmetries are what makes the world interesting. (Symmetry is broken)
  • In physics, as in art, simplicity alone is not necessarily the highest goal. Life and the universe are rarely perfect, and almost all symmetries are broken.
    • The goal is to make theories that are richer and sometimes even more beautiful without compromising their elegance.
    • The concept of the Higgs mechanism, which relies on the phenomenon of spontaneous symmetry breaking.
    • Scottish physicist Peter Higgs, lest the Standard Model particles acquire mass.
    • Without the Higgs mechanism, all elementary particles would have to be massless.
  • Spontaneously Broken Symmetry:
    • Spontaneous symmetry braking is not only ubiquitous in physics, but is a prevalent feature of everyday life.
      • occurs when all physical laws a symmetry but the actual physical system does not. Spontaneously broken symmetries are symmetrices that are preserved at high energyies but broken at low energies. The weak force symmetry is spontaneously broken.
    • E.g.: a number of people are seated around a circular table with water glasses placed between them. Which glass should someone use?
      • But one has to be chosen, and after that, left and right are no longer the same in that there is no longer a symmetry that interchanges the two.
  • >Top Higgs Mechanism:
    • Each type of field generates its own particular type of particle.
      • An electron filed is the source of electrons. Similarly, a Higgs field is the source of Higgs particles.
      • As with heavy quarks and leptons, Higgs particles are so heavy that they aren't found in ordinary matter.
      • Higgs particles are too heavy to have been produced with the energies that experiments have explored so far.
        • Higgs will be created within a few years, when the highter-energy LHC collider comes into operation.
    • A non-zero Higgs field distributes weak charge throughout the universe. It is as if the nonzero weak-charge-carrying Higgs field paints weak charge throughout space.
      • The vacuum itself carries weak charge when one of the two Higgs fields takes a nonzero value.
      • Weak gauge boson interact with this weak charge of the vacuum, just as they do with all weak charge.
      • The Higgs field plays a role very similar to that of the traffic cop in the story, restricting the weak force's influence to very sort distances.
    • The weak charges in the vacuum have a density that corresponds roughly to charges that are separated by $10^{-15}$ cm (1 femto).
      • With this weak charge density, the masses of the weak gauge bosons ($W^-, W^+, Z$) take their measured values of approximately 100 GeV (126 GeV as the Standard Model)
      • Quarks and leptons interact with the Higgs field distributed throughout space and are therefore also hindered by the universe's weak charge.
    • Photon is the only gauge boson involved in the electroweak force that is impervious to the weak charge of the vacuum.
      • The chief distinguishing feature of the photon is that it travels unfettered through the weakly charged vacuum and therefore has no mass.

10. 素粒子質量の起源:

>Top11. Scaling and Grand Unification: Relating interactions at different lengths and energies.

  • Grand Unified Theory (GUT), in which the three nongravitational forces merge into a single unified force at high energies.
    • the unification energy is about 1000 trillion GeV, and the Planck scale energy, where gravity gets strong, is about a 1000 times greater than that.
  • Zooming in and out:
    • One advantage of an effective field theory is that even if you don't know what interactions take place over short distances, you can still study the quantities that matter at the scales that interest you.
      • When you mix paint, you don't need to know its detailed molecular structure. But you probably want to know the properties that you readily perceive, like color and texture. However, if you did know your paint's chemical composition, the rules of physics would allow you to deduct some of those properties.
      • Similarly, if you don' t know the short-distance (high energy) theory, you won't be able to derive measurable quantities. However, when you do know the short-distance details, quantum field theory tell you precisely how to relate the different effective theories that apply to different energies.
  • 1974 Renomalization group; by Kenneth Wilson;
    • When we curled up dimensions, we ignored all the details of what happened inside the extra dimensions and assumed that everything could be described in lower-dimensional terms.
      • That way, you can neglect heavy particles that you'll never produce and short-distance interactions that will never occur. Instead, you can focus your calculations on particles and interactions that are relevant at the energy you can achieve.
      • However, if you do know the more precise theory that applies at smaller distances, you can use that information to calculate quantities in the effective theory that interests you.
  • >Top Virtual Particles:
    • They pop in and out of existence, lasting only the barest moment.
    • A particle moving very fast clearly carries a lot of energy. A virtual particle can have enormous speed but no energy. Virtual particle are a strange feature of quantum field theory that you have to include to make the right predictions.
    • The uncertainty principle tells us that it would take infinitely long to measure energy (or mass) with infinite precision, and that the longer a particle lasts, the more accurate our measurement of its energy can be.
    • Vacuum as a reservoir of energy - virtual particles are particles that emerge from the vacuum, temporarily borrowing some of its energy. They exist only fleetingly and then disappear back into the vacuum, taking with them the energy they borrowed.
    • All particle-antiparticle pairs can in principle be produced, albeit only for very short visits, too short to be seen directly.
    • A real physical photon can turn into a virtual electron and a virtual positron, which can then turn back into a photon. (>Fig.)
  • Why interaction strength depends on distance:
    • >Top Anarchic principle:
      • follows from quantum mechanics, which tells us that all particle interactions that can happen will happen. In quantum field theory everything
    • Physical particles can turn into virtual particles, which can interact with each other and then turn back into other physical particles. In such a process, the original physical particles might reemerge or they might turn into different physical particles.
      • Even though the virtual particles wouldn't last long enough for us to observe them directly, they would nonetheless affect the way real observable particles interacted with one another.
      • Not only direct interactions between physical particles, but also indirect interactions - those that involve virtual particles - play a role in communicating forces.
  • Virtual correction to electron-positron scattering: (>Fig.)
    • An electron and a positron annihilate into a photon, which in turn splits into a virtual electron-positron pair which then annihilate back into a photon, which in turn coverts to an electron and positron. The intermediate virtual electron and positron thereby affect the strength of the electromagnetic force.
  • Grand Unification Theory (GUT):
    • 1974 Georgi-Glashow:
    • "We present a series of hypotheses and speculations leading inescapably to the conclusion ... that all elementary particle forces (strong, weak, and electromagnetic) are different manifestations of the same fundamental interaction involving a single coupling strength."
    • GUT predicts that the proton decays. After a very long time, they would decay.
      • a single high-energy force turns into the three known nongravitational forces at low energies. For the three forces to unify they must have the same strength at high energies.
      • The net number of quarks in the universe would not remain the same, and a quark could change into a lepton, making the proton decay.
      • the decay rate of the proton is very slow - the lifetime would far exceed the age of the universe.
      • All familiar matter would ultimately be unstable.

11. スケーリングと大統一:

  • Virtual Particle:


  • Virtual correction to electron-positorn scattering:



>Top 12. The Hierarchy Problem; the only effective trickle-down theory:

  • the Standard Model predictions:
    • The masses and charges associated with the electromagnetic, weak, and strong forces have been tested to incredbly high accuracy.
      • Experiments at the colliders at CERN, SLAC (Stanford), Fermilab (Illinois) have all confirmed with exquiste precision the Standard Model predictions.
      • the Higgs mechanism explains how the vacuum breaks electroweak symmetry and gives masses to the W and Z gauge bosons, as well as the quarks and the leptons.
    • >Top The hierarchy problem in a GUT:
      • the masses are not at all the same; even those particles that experience similar forces must have enormously different masses; not by a factor of ten, but of ten trillion.
      • the new particle in GUT has to be extremely heavy.
      • in a GUT; the weak force and the strong force should be interchangeable at high energy.
      • the strongly charged particle that is partnered with the Higgs particle can interact simultaneously with a quark and a lepton and thereby enable the proton to deay.
        • To avoid too rapid a decay, the strongly interacting particle must be extremely heavy; about one million billion GeV.

12. 階層問題:

  • A contribution to the Higgs particle's mass:


  • The hierarchy problem:
Energy   Length
$10^{27}$GeV Planck scale  
$10^{18}$GeV $10^{-33}$cm
$10^{15}$GeV 16 orders of magnitude $10^{-30}$cm
: :
weak scale $10^{-15}$cm
GeV proton mass $10^{-12}$cm
electron mass $10^{-9}$cm

>Top 13. Supersymmetry: A leap beyond the Standard Model

  • "Super" words abound in physics; superconducting, supercooling, supersaturated, superfluid, Super conducting Supercollider (SSC)
    • Supersymmetry was truly surprising.
    • Supersymmetry as a symmetry of nature is still hypothetical.
    • In a supersymmetric world every known particle is paired with another (superpartner), with which it is interchanged by a supersymmetry transformation.
      • Fermions have half integer spin, while bosons have integer spin.
      • such asymmetry defies logic. Symmetry transformation are supposed to leave system unchanged. But supersymmetry transformations interchange particles that are manifestly different; fermions and bosons.
      • Superpartners, should they exist, must be more massive than their Standard Model partners. Because high-energy colliders might not yet have had enough energy to produce them. This would explain why we have not yet seen them.
      • Once supersymmetry is broken, flavor-changing interactions can occur. These are processes that change quarks or leptons into quarks or leptons of another generation with the same charges.
        • These are very strange processes; they change the identity of known particles, and they occur only rarely in nature. But most theories of broken supersymmetry predict that they should occur very often.
  • Superhistory:
    • 1971: Pierre Ramond; put forward the first supersymmetric theory; contained 2D supersymmetry and evolved into the fermionic string theory.
    • 1973: Julius Weess, Bruno Zumino; developed 4D supersymmetric theory.
    • Dmitri Volkov and Vladimir Ocelot in USSR independently derived another 4D supersymmetric theory.
    • 1974: Sergio Ferrara and Bruno Zumino; formalism of superspace; but did not yet include gravity.
    • 1976; Sergio Ferrara, Dan Freedman, and Peter van Nieuwenhuizen; solved by constructing supergravity
    • Ferdinando Gliozzi, Joel Scherk, and David Olive; discovered a stable string theory. as an fermionic string theory.

13. 超対称性:

  • Supersysmmetric partners:
Particle Supeerpartner
lepton slepton
electron selectron
quark squark
top stop
gauge boson gaugio
photon photino
W bozon wino
Z bozon zino
gluon gluino
graviton gravitino


>Top 14. Allegro (Ma Non Troppo) Passage fro Strings:

  • Incipient Unrest:
    • Quantum mechanics and general relativity peacefully coexist over a wide range of distances.
      • These two very different theories never adequately negotiated the extremely tiny distance as the Planck scale $10^{-33}$ cm.
      • The gravitational force law tells us that on even tinier scales, the force of gravity is enormous; at the Planck scale length, gravity cannot be ignored.
      • >Top At the Planck scale length, instead of a gradually undulating geometry there should be wild fluctuations and loops and handles of spacetime branching off.
        • Given the conflict between them, there is no choice but to bring in an external arbiter as an alternative to both.
        • The graviton is the only known massless particle whose spin is 2 - not 1 as for other gauge bosons, or $\frac{1}{2}$ as for quarks and leptons.
        • Current theories of cosmology conjecture that the universe began as a tiny ball; the bang of the Big Bang.
        • Black hole's horizon and at the singularity, the place in the center of the black hole.
  • String training:
    • Strings - vibrating, 1D loops or segments of energy.
    • We now believe that string theory can contain different, independent types of string, each of which can oscillate in many possible ways
      • strings are 1D objects.
      • two type of string; open strings, having two endpoints, and closed strings, are loops with no ends.
      • For both open and closed strings, the resonant modes are those that oscillate an integer number of times along the string's length; any other waves - those that don't complete an integer number of oscillations won't occur.
        • the string oscillates determines all of a particle's properties, such as its mass, spin, and charge.
        • Known particles, which are relatively light, arise from strings with the fewest oscillations.
        • a mode with no oscillations could be a familiar light particle, such as ordinary quark or lepton.
  • >Top Superstring Theory:
    • it contained spin $-\frac{1}{2}$ particles, giving it the potential to describe the Standard Model fermions such as electron and quarks.
      • it did not contain the tachyon that had plagued the original string theory.
      • the spin-2 particle behaved just as a graviton should.
  • Superstring Revolution:
    • Strings could conceivably move around in 3, r, or more dimensions. Calculations indicated that the correct number (including time) was 10.
    • 1969: Steven Adler, et al, showed that even when a classical theory preserves a symmetry, quantum mechanical processes involving virtual particles sometimes violate that symmetry; as called anomalies.
      • Virtual quarks and leptons would lead to anomalous quantum contributions that would break the symmetries of the Standard Model.
      • However, the sum of the quantum contributions from the quarks and the leptons adds up to zero.
    • 1980: Schwarz and Michael Green; worked out the subsequences of the superstring; known as anomalies.
      • String theory was largely ignored until 1984, the year that Green and Schwarz demonstrated a starling feature of the superstring which convinced many other physicists that they were on a promising track.
    • 1985:
      • Princeton collaboration of David Gross, et al. derived a theory of heterotic string;
      • In string theory, a vibrational mode can move either clockwise or counterclockwise along the string. "Heterotic" was used because waves moving to the left were treated differently from those waves moving to the right.
    • 1985, Philip Candelas, et al.; complicated way of curl up the extra dimensions, namely a competitiveness known as Calabi-Yan manifolds.
    • >Top 1980s: Theory of Everything (ToE);
      • String theory was more ambitious even than Grand Unified Theories; with string theory, physicists hoped to unify alll forces (including gravity) at an energy higher even tthan the energy associate with GUTs.
  • Aftermath of the Revolution:
    • Even in the absence of particles, the universe can carry energy known as vacuum energy.
      • Positive vacuume energy accelerated the expansion of the universe, while negaive energy makes it collapse.
        • Einstein first proposed such an energy in 1917 in order to find a static solution to his equations of general relativity.
        • Astronomers have recently measured the vacuum energy in our cosmos (dark energy or the cosmological constant) and found a small positive value.
        • The supernova measurements tells us that the expansion of the universe is accelerating.
    • String theory has yet to explain why the universe's vacuum energy is as small as we know it must be.
      • Nonetheless, string theory is a remarkable theory. It has already led to important insights into gravity, dimensions, and quantum field theory and it's the best candidate we know of for a consistent theory of quantum gravity.
      • Model builders are still waiting for the experimental clues that tell us which of the myriad possibilities correctly describe physics beyond the Standard Model; energy higher than a TeV.

14. 急速な紐の通路:

  • String oscillation models:
    Open string and Closed string
  • openstring
  • closedstring
  • String Theory:
  • Superstring theory:
  • Spin-2 particle=graviton?

>Top 15. Supporting Passages; Brane development

  • Branes:
    • String theory is no longer just the theory of strings extending in one spatial direcrion, but also the theory of ranes that can extend in 2, 3, or more dimensions.
    • We now knw that branes, which can extend in any number of dimensions up to the number that supersting theory contains, are just as much a part of superstirng theory as are strings themslves.
  • >Top M-theory: began in 1980s and developed in 1990s.
    • an 11-dimensional theory that embraces both superstring theory and 11D supergravity.
    • M stands for membrane, magic, mystery, Missing theory.
  • Duality:
    • is one of the most exicitng conecpts of the last ten years in particle physics and string theory.
    • in 1977, Claus Montonen, et al. showed that a particular theory remained exactly the same if the particles and strings of the theory were interchanged.
  • >Top Perturbation theory;
    • lest you creep up on the answer to a question in the weaky interacting theory by starting from the thoery with no interactions and calculatin small improvements in incremental stage.; to refine a claculation in successive steps until you reach any desired level of precision.
  • More on duality:
    • 10D supergravity does not contain strings, whereas 11D supergravity does not. Even though 11D supergravity does not contain strings, it does contain 2-branes. But ulike strings, which have only one spatial dimension, 2-branes have two.
      • Now, suppose that one of the 11D is rolled up into an extremely tiny circle; 2-brane that encircles the rolld-up circular dimension looks just like a string.
      • This means 11D supergravity theory with a rolled-up dimension appears to contain strings.
      • A rolled-up dimension is invisible at long distances or low energies. (>Fig.)

15. 脇道の通路:

  • Rolled-up brane:


>Top 16. Bustling Passages; Braneworlds

  • Particles, Strings, and Branes:
    • A string that begins and ends on a single brane can give rise to a gauge boson.
    • A string with each end on a different brane gives rise to a new type of gauge boson. When the brans are separated, the gauge boson has nonzero mass.
      • the mass of the particles arising from the vibrational modes of this string grows with the distance between the branes. This mass is like the energy that gets stored when you stretch a spring.
    • Now imagine that many branes are superimposed. there would then be many new types of open string because the two string ends can be confined to any of the branes.
      • Open strings that extend between different branes, or the strings that begin or end on any single one of the branes, imply new particles, composed of the vibrational modes of these strings. (>Fig.)
      • Particles on separated branes don't interact with each other directly. Interactions are local; they can take place only among particles in the same place. particles on separated branes would be too far apart to interact with each other directly.
      • E.g.: a huge tennis stadium with separate matches going on throughout. Each ball would stay on it own isolated court.
  • Gravity:
    • Gravity, unlike all other forces, is never confined to a brane.; graviton is a mode of a closed string.
      • Closed strings have no ends, and therefore there are no ends to pin down on a brane.
      • Graviton, unlike gauge bosons or fermions, must travel though the entire higher dimensional spacetime. there is no way to confine gravity to lower dimensions.
  • Model Braneworlds:
    • Brane scenarios introduced many new possibilities for the global nature of spacetime.
    • As of now, string theory doesn't tell us whether brans exist and, if they do, how many there are. We know only that branes are an essential theoretical piece of string theory.
    • If Standard Model particles are confined to a brane, then we are as well, since we and the cosmos that surrounds us are composed of these particles.
      • There might therefore be entirely new and unfamiliar particles that experience different forces and interactions from the ones we know.
      • The particles and forces we observe might be only a small part of a much larger universe.
  • >Top Hořava-Witten Theory:
    • " branes with 9 spatial dimensions are separated along the 11th spacetime dimension (10th spatial dimension). The bulk includes all spatial dimensions; those 9 that extend in the spatial directions along the two branes, and the additional one that extends between them. (>Fig.)
      • 11th dimension is not rolled up, but is instead bounded between two branes.
      • If the HW braneworld is to correspond to reality, size of tis dimension must be unseen. HW assumed that six dimensions were curled up into a tiny Calabi-Yau shape.
      • it can accommodate not only the Standard Model particle and forces, but also a full GUT. Because gravity originals in higher dimensions, it's possible for gravity and other forces to have the same strength at high energy in this model.
    • Why braneworlds can matter for real-world physics.
      1. More than a single brane: can contain forces and particles that interact with each other only weakly because of the distance between the two branes.
      2. Any braneworld introduces new length scales into physics; like the size of the additional dimensions, might be relevant to unification or the hierarchy problem.
      3. Branes and the bulk can carry energy.; it doesn't depend on the particles that are present.

16. 賑やかな通路:

  • Strings on the brane:


  • Hořava-Witten Theory:


>Top 17. Sparsely populated passages; multiverses and sequestering:

  • Sequestering; extra dimensions could prove to be important for particle physics.
    • sequestering might also be the reason that the anarchic principle, which says that everything should interact, doesn't always hold true.
      • If particles are separated in extra dimensions, they are less likely to interact with one another.
        • Extra dimensions introduce a way to separate particles.
      • Whereas 4D theories face serious problems because supersymmetry-breaking models generally introduce unwanted interactions, sequestered supersymmetry-breaking models appear to be far more promising.
        • Sequestering might also explain why particles have different masses from one another, and why proton decay does not occur in extra-dimensional models.
    • "Natural" : for physicists, it is only natural to expect the expected.
      • Symmetries essentially provide an extra rule about which interactions can conceivably happen.
      • Suppose that you prepare six table setting; permit a symmetry transformation that interchanges every pair of setting; all six people have the same number of forks, knives, spoons, and chopsticks.
    • Particles on different branes;
      • don't interact strongly because interactions are always local; only if there are interacting particles that can travel from one brane to the other.
      • This suppression of interactions between particles sequestered in different places would be similar to the suppression of International information in a country. (xenophobia)

17. 過疎の通路:

  • Superesymmetry breaking:


>Top 18. Leaky passages; Fingerprints of extra dimensions

  • Kaluza-Klein particles:
    • New particles that originatd in extra dimensions, but appear to us as extra particles in our 4D spacetime, are Kaluza-Klein particles.
    • The relationship between mass and momentum imposed by special relativity tells us that extra-dimensional momentum would be seen in 4D world as mass.
  • Determining Kaluza-Klein Masses:
    • The extra-dimensional momenta that the KK particles carry would appear to us in our apparently 4D word as a distinctive pattern of KK particle masses.
    • Quantum mechanics associates all particles with waves, and only those waves that can oscillate an integer number of times over the extra-dimensional circle are allowed.
      • The lightest KK particle is therefore the one associate with this constant probability value in the extra dimension.
      • Heavier KK particles, which carry nonzero extra-dimensional momenta, will be the first real evidence of extra dimensions.
      • The wavelength is determined by the size of thee extra dimension's circumference.
      • the lighter KK particle that "remembers" its extra-dimensional origin.
    • Kaluza-Klein particles correspond to the waves that oscillate an integer number of times around the curled-up dimension.
      • Waves with more oscillations correspond to heavier particles.
      • KK particles and their masses could tell us quite a lot about extra-dimensional properties. (extra dimensions' sizes and shapes)
      • KK particles resemble the multiple generations of an immigrant family; Younger generation born in US fully assimilate American culture, and don't betray their foreign roots.
  • Experimental Constraints:
    • Until recently, most string theorists assumed that extra dimensions are no bigger than the minuscule Planck scale length ($10^{-33}$ cm)
    • Perhaps extra dimensions are bigger and KK particles are lighter. Why not ask instead what experimental tests tell us about an extra dimension's size?
    • So far there has not been any sign of such charged particles at colliders. (about TeV scale about $10^{-17}$ cm)
    • Current experimental constraints tell us that extra dimensions cannot be any larger than $10^{-17}$ cm. This is extremely small, about 10 times smaller than the weak scale length.
      • But it is huge compared with the Planck scale length, 16 orders of magnitude small.
      • This means that extra dimensions could be much bigger than the Planck scale length and sill have evaded detection.

18. 漏れる通路:

  • Kaluza-Klein theory:


  • Kaluza-Klein praticles:


>Top 19. Voluminous Passages; Large extra dimensions

  • 1998: Arkani-Hamed, Dimopoulos, and Dvali (ADD):
    • How very large dimensions might explain the weakness of gravity.
    • ADD wrote: extra dimensions could be as big as a millimeter (or even 10 times smaller).
    • Branes can trap quarks, leptons, and gauge bosons so that only gravity experiences the full higher dimensionality of space.
      • Extra dimensions can be so large that they can explain why gravity is so much weaker than the electromagnetic, weak, and strong forces.
      • In the ADD scenario, which assumes that everything other than gravity is confined to a brane, everything that doesn't involve gravity wold look exactly the same as it would without the extra dimensions, even if the extra dimensions were extremely large.
      • Your eye detects photons, and photons in the AD model are trapped on a brane. Therefore all objects you see would look as if there were only three spatial dimension.
      • In the ADD model, the only particle that must have KK partners is the graviton, which must travel in the higher dimensional bulk.
    • Until recently, the main question that why the weak scale mass is so small, despite the large (Planck-scale-mass-size) virtual contributions to the Higgs particle's mass that tend to make it larger.
      • They proposed that the fundamental mass scale that determines gravity's strength is not the Planck scale mass, but a much small mass scale, close to a TeV.
    • After all, the reason that the Plank scale mass is so big is that gravity is weak - gravity's strength is inversely proportional to this scale. A much smaller fundamental mass scale for gravity would make gravitational interactions far too strong.
  • Back to the hierarchy problem:
    • Suppose that gravity in a higher dimensional theory doe not depend on the enormous Plank scale mass of $10^{19}$ GeV, but instead on a much smaller energy, about a TeV, 16 orders of magnitude smaller.
      • They chose a TeV to eliminate the hierarchy problems; if a TeV or some nearby energy were the energy at which gravity became strong, there would be no hierarchy of masses in particle physics. Everything would be characterized by the TeV scale. So maintaining a reasonably light Higgs particle with mass of about a TeV would not be a problem in their model.
  • Collider searches for large extra dimensions:
    • If the large-dimension proposal applies to the real world, the graviton KK partners would be light enough to be produced at accelerators, no matter how many extra dimensions there were.
      • Current colliders create more than enough energy to make such low-mass particles.
      • If there were two extra dimensions, about 100 billion trillion KK modes would be light enough to be produced at a collider operation at an energy of about a TeV.
      • The rate of producing at least one of these particles would be fairly high, even in the rate of producing any single one of them were extremely low.
        • It would be as if someone hinted something to you in such a subtle manner that you didn't take it to heart the first time you heard it. But afterwards, 50 people repeated the same thing. Even though you wouldn't take much notice the first time you heard the messaged, by the 50th time the message would register.
      • The large Hadron Collider, which will study TeV-scale energies, could produce KK particles at a measurable rate if the ADD idea is correct.
    • KK particle production in the ADD model (>Fig.)
      • Protons collide, and a quark ad an antiquark annihilate into a virtual gluon. The virtual gluon turns into an undetected KK particle and an observable jet. The gray lines are sprays of additional particles that protons always emit when they collide.

19. たっぷりした通路:

  • KK particle production in the ADD model:


>Top 20. Warped Passage: A solution to the hierarchy problem

  • The geometry that we'll consider here contains 2 branes that bound a 5D of space (>Fig.)
    • The universe has 5 spacetime dimensions, but the Standard Model resides on a brane (the Weakbrane) that has four.
      • The total number of spacetime dimensions.
      • There are 2 btanes with a 5th dimension that etends between them.
      • The particles and the distribution of energy are different, and the theory is not supersymmtric.
        • We assue that all of the Standard Model particles, along with a Higgs particle responsible for breaking electroweak symmetry, are confined to one of the two branes.
        • W'll assume that gravity is the only forrce that exists throughout the 5th dimension.
        • Each of the branes would look like a conventional 4D universe. Gauge bosons and particles confined to the branes would communicate forces and interact as if the 5D didn't exist.
        • Gravity is different since it is not restricted to a brane, ubt instead exists in the full 5D bulk. But this does not necessarily mena that is is felt equally everywhere.
        • Energy o nbt ebranes and int 5D bulk curves spacetime, and this makes an enourmous difference to the gravitational field.
    • The 5D spacetime is nonetheless curved.
      • The warp factor is a function that changes the overall scale for position, time, mass, and energy at each point in the 5D.
      • A curvd space iwth flat slices is pictured in Fig. It is a filled-in funnel.
        • Ths curvature would be reflectd in an overall rescaling of the measuring rode of space and the clock spd for time, which would be different at each point in 5D.
        • The graviton si shte particle that communicates the gravitational force, and its probaility function tells us the likelihood of finding the graviton at nay fixed position in space.
        • The graviton's probability unction alls of exponentially as it move away from the Gravitybrane and twards the Weakbrane.
        • The distance between branes in this warped geometry need only be a little larger than the Planck scale length.

20. ワープした通路:

  • Warped 5D geometry with 2 branes:


  • A filled-in funnel:


>Top 21. Warped Annotated "Alice":

  • The tittle borrows from Martin Gardner's delightful Annotated Alice, in which he explains the wordplay, math riddles, and references in Lewis Carroll's Alice in Wonderland and Through the Looking Glass.
  • The brane itself is large and flat and has only three spatial dimensions. Only gravity makes contact with the additional dimension.
    • Remember that 5D space had 4 spatial dimensions and one of time, whereas the brane has 3 spatial dimensions. I'll still call time the 4th dimension, and I'll call the additional dimension the 5th.
    • The Fat Cat, unlike Branesville residents, is not confined to the brane.
    • Everything is bigger and lighter near the Weakbrane. Athena's shadow over Branesville grew as she got close to the Weakbrane and further away from the Gravitybrane.
    • The 5th dimension does not have to be very big in order to solve the hierarchy problem.
    • Gravity is feeble on the Weakbrane, where the graviton's probability function is so small.
    • On the Gravitybrane, gravity is no weaker than the other forces.
    • The petulant graviton is complaining that on the Weakbrane, gravity is much weaker than the electromagnetic, weak, and strong forces. Gravity would be much stronger closer to the Gravitybrane.
    • Things are bigger and time is slower on the Weakbrane. the rabbit's laxness is accounted for by rescaling time.

21. ワープした注釈付きのアリス:

>Top 22. Profound Passage; An infinite extra dimension:

  • The localized graviton:
    • the localized graviton has unrestricted access to an infinite 5D. But it is nonetheless highly concentrated in the vicinity of a brane, and has very low probability of being found far away.
      • Because the graviton only rarely ravels outside a limited region, the extra dimension can be infinite without giving rise to any dangerous effects that would rule the theory out.
  • Kaluza-Klein Partners of the Graviton:
    • Graviton travels almost exclusively along the brane and has only a tiny probability of leaking out into 5D.
      • From the graviton's perspective, space looks as if the 5D is on $10^{-33}$ cm in size, rather than of infinite extent.
      • In localized gravity, the massless KK particle is the localized graviton. It is concentrated close to the Gravitybrane.
      • All other KK particles are concentrated far from the Gravitybrane; the shape of their probability function and the locations where they peak depend on their mass.

22. 深淵な通路:

  • Graviton's probability function in infinite warped spacetime with a single brane:


>Top 23. A Reflective and Expansive Passage:

  • Reflections:
    • After our theory was accepted and no longer thought incorrect, some physicists actually went overboard in a different direction, claiming our theory was nothing new.
    • Localized gravity turned out to have strong overlaps with the most important string develpments of the time; both our work and the research of string theorists involved a similarly warped geometry.
  • Locally Localized Gravity:
    • how many dimensions of space are there?
    • You now know that extra dimensions can be hidden either because there are curled up and small, or because spacetime is warped and gravity so concentrated in a small region that even an infinite dimension is invisible. Either way, whether dimensions are compact or localized, spacetime would appear to be 4D everwhere, no matter where you are.
    • Gravity acts as it does in 4D if you're near the brane.
      • Although the graviton's probability function is largest on the Gravitybrane, objects everywhere can interact with one another by exchanging a graviton, and therefore all objects would experience 4D gravity, independently of location.
      • Gravity everywhere looks 4D because the graviton's probability function is never actually zero - it continues on for ever.

23. 収縮・膨張する通路:

>Top 24. Extra Dimensions; are you in or out?

  • Warped geometry and Duality:
    • how can a 4D and a 5D (or 10D) theory have the same physical implications?
    • The answer is that an object moving through 5D would appear in the dual 4D theory as an object that grows or shrinks. this is just like
    • Athena's shadow on the Gravitybrane, which grew as she move away from the Gravitybrane across the 5D. Furthermore, object moving past each other along the 5D correspond to objects that grow and shrink and overlap in 4D.
  • T-duality:
    • it exchanges a space with a tiny rolled-up dimension for another space with a huge rolled-up dimension. Odd as it may seem, in string theory, extremely small and extremely large rolled-up dimensions yield the same physical consequences.
  • Mirror Symmetry:
    • T-duality applies when a dimension is rolled up into a circle. But an even weirder symmetry than T-duality is mirror symmetry, which sometimes applies in string theory when 6D are rolled up into a Calabi-Yau maniforld.
    • Mirror symmetry says that 6D can be curled up into two very different Calabi-Yau manifolds, yet the resulting 4D ling-distance theory can be the same.
  • Matrix Theory:
    • is a tool for studying string theory, provides still more mysterious clues about dimensions.
    • Superficially, it looks like a quantum mechanical theory that describes the behavior and interactions of Do-branes (pointlike branes) moving through 10D.
    • Furthermore, the theory of Do-branes mimics supergravity in 11D, not 10D.
  • What to think?:
    • No one knows what the best description would be when a dimension is neither very big nor very small. Perhaps our notion of spacetime breaks down altogether once we try to describe something so small.
    • We know from quantum mechanics that it takes a lot of energy to investigate small length scales. But once you put too much energy into a region as small as the Planck scale length, you get a black hole; then not way to know what's happening inside.

24. 余剰次元:

>Top 25. Conclusion:

  • We don't yet know how to harness the force of gravity or teleport objects across space.
    • We don't know how to connect universes in which you could loop through time to the one in which we live; no one can create a time machine.
    • Our goal is to learn how its pieces fit together and how they've evolved into their current state. What at the connections that we haven't yet figured out?
    • Even if we have yet to understand the ultimate origin of matter at the deepest level.
    • Even if we don't know the most basic elements of spacetime, we do understand its properties for distances far away from the Planck scale length.
  • Physicists are now developing detailed theories of black holes in extra-dimensional worlds.
    • Cosmological observations might also ultimated tell us more about the structure o of spacetime.
    • If we live in a higher-dimensional universe, it must have been very different earlier on.
    • We might learn about dark matter hidden on other branes, or cosmological energy stored by hidden higher dimensional objects.
  • the Large Hadron Collider (LHC) at CERN will turn on and probe physical regions no one has ever observed before.
    • LHC will have enough energy to produce the new types of particle; superpartners or other particles that 4D models predict.; also be Kaluza-Klein particles.
  • The universe is about to be pried open; Astrophysical observations; Discoveries at the LHC.

25. 結論:

  • a

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