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The historic achieve-ments brought forth by physics in the 20th century:

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Man discovered, for the first time since our ancestors discovered fire, the second and the vastly stronger source of energy: nuclear power.

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Man learned to manipu-late electrons to create the transistor which led to the modern computer, there-by greatly increasing human productivity.

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Man learned how to probe into structures of atomic dimensions which led to the double-helix, thereby ushering in bioengineer-ing technology.

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Man take first steps on the moon.

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However, from the viewpoint of physicists, the most important advances are the profound revolutions in our understanding of the basic concepts of physics.

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Space

Time

Motion

Energy

Force

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There were three themes that, singly and together, underlie the chief new ideas in the 20th century physics. We may call them:

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Thematic Melodies of Twentieth Century Theoretical Physics:

QuantizationSymmetry

Phase Factor

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1. Quantization

11Max Planck (1858-1947)

12

13

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1900 Planck

1905 Einstein

1913 Bohr

Quantization

15Albert Einstein (1879-1955)

16Niels Bohr (1885-1962)

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It was the spring of hope,

it was the winter of despair

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At present I am myself most optimistic as regards the future of the theory.

Bohr to Rutherford 1918

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Physics is once again at a dead end at this time. For me, at any rate. It is much too difficult.

Pauli to Kronig,May 21, 1925

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Heisenberg’s mechanics has restored my zest for life.

Pauli to Kronig,October 9, 1925

21Wolftgang Pauli (1900-1958)

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Do not enter into this conflict, we are both much too kind and gentle to participate in that kind of struggle. Both Bohr and Heisenberg are tough, hard nosed, uncompromising and

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indefatigable. We would just be crushed in that juggernaut.

Kramers to Klein 1927Quoted in Pais’ <Genius of

Science>, p.159 (2000)

24J.R. Oppenheimer (1904-1967)

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It was a period of patient work in the laboratory, of crucial experiments and daring action, of many false starts and many untenable conjectures. It was a time of earnest correspondence and

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hurried conferences, of debate, criticism, and brilliant mathematical improvisation.

For those who partici-pated, it was a time of creation; there was terror as well as exaltation in their new insight . It will probably not

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be recorded very completely as history. As history, its recreation would call for an art as high as the story of Oedipus or the story of Cromwell, yet in a realm of action so remote from our common experience that

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it is unlikely to be known to any poet or any historian.”

J.R. Oppenheimer Reith Lectures 1953

29Werner Heisenberg (1901-1976)

30P.A.M. Dirac (1902-1984)

31Erwin Schrödinger (1887-1961)

32Enrico Fermi (1901-1954)

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Pauli — Power

Fermi — Solidity, Strength

Heisenberg — Deep Insight

Dirac — Cartesian Purity

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2. Symmetry

(= invariance)

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The five regular solids with maximum symmetry. Reprinted from A.V. Shubnikov and V.A. Koptsik, Symmetry in Science and Art (Plenum, 1974).

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1905 Einstein

1908 Minkowski

Symmetry

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“Superfluous learnedness”

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… that the basic demand of the special theory of relativity (invariance of the laws under Lorentz-transformations) is too narrow, i.e. that an invariance of the laws must be postulated also relative to non-linear transformations of

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the coordinates in the four-dimensional continuum.

This happened in 1908.

Einstein: Autobiographical Notes

in <Albert Einstein>, ed. P.A. Schilpp, p.67

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With the introduction of quantum mechanics in 1925, symmetry became very important. The mathematical language for symmetry is groups.

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It has been rumored that the “group pest” is gradually being cut out of quantum physics.

H. Weyl, Nov. 1930

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Symmetry gradually became the thematic melody (1927-1970)

atomic, molecular physics

nuclear physics

elementary particle physics

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A great shock created by Prof. C. S. Wu in 1957

Parity Nonconservationin Weak Interactions

44C.S. Wu (1912-1997)

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Now, where shall I start? It is good that I did not make a bet. It would have resulted in a heavy loss of money (which I cannot afford); I did make a fool of myself, however (which I think I can afford)

Pauli 1957

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Never before or afterward have I seen him so excited about physics.

Heisenberg 1978

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3. Phase Factor

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So if one asks what is the main feature of quantum mechanics, I feel inclined now to say that it is not non-commutative algebra, it is the phase.

Dirac 1972

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phase factor =

　　

ie

360to0

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Phase factor became particularly important through the proposal of Weyl in 1918.

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Weyl introduced

gauge factor =

　　

e

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Then London and Fock added so that gauge → phase

,1i

iee

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1918 Weyl1927 Fock & London

*****

Gauge Theory

Flexibility of phase factor→ electromagnetic equation

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Gauge invariance is

a very large symmetry.

　

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E i n s t e i n 1 9 2 4

W e y l 1 9 1 8

S c h r ö d i n g e r 1 9 2 2

d e B r o g l i e 1 9 2 3

B o s e 1 9 2 4

S c h r ö d i n g e r 1 9 2 6

F o c k 1 9 2 7

L o n d o n 1 9 2 7

W e y l 1 9 2 9

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1918 Weyl:

dxA

eexp

Stretch Factor1922 Schrödinger: Bohr orbit, exponent

h

n

“Remarkable Property”******

stretch factor = 1

Phase Factor

i

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The de Broglie interpretation of the quantum rules seems to me to be related in some ways to my note in the Zs. F. Phys. 12, 13, 1922,….. The mathematical situation is, as far as I can see, the same,

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only from me much more formal, less elegant and not really shown generally.

Schrödinger to Einstein, Nov 3, 1925

59Erwin Schrödinger (1887-1961)

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The three thematic melodies were introduced in the first half of the century, their developments in the next half century were:

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Developments

Variations

Intertwinings

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Motives:

(1) Discovery of more andmore “strange” particlesneed a general principlefor interactions throughsymmetry.

Generalization of Gauge Symmetry

eBpeAp

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　　(2) Conservation of charge

→ electromagnetic field

Conservation of energy 　 → gravitational field

Why other conservation laws do not → specific field?

　

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(3) Some Conservation laws were related to global gauge transformation. This is not consistent with the spirit of the concept of localized fields.

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Principle for Interactions

Non-Abelian Gauge Theory

Conservation Law

LocalizedGauge

Symmetry

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Mathematical Language of Symmetry: Groups

Galois (1811-1832)Lie (1842-1899)

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Simplest Lie Group is Phase Factors

Non-Abelian Lie Groups are Generalizations of this Phase Factor

ie

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QM phase1926

Flexibility in Definition of Phase

1929EM is Gauge Theory

Flexibility Generalized

1954Non-Abelian Gauge Theory

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Usual Symmetry Gauge Symmetry

Equation Equation

Sol. Sol. Sol. Sol. Sol. Sol.

(Different State)

(Same State)

Schematic diagram illustrating the difference between usual symmetry and gauge symmetry. The horizontal arrows represent symmetry transformations which relate the solutions (sol. in the diagram). For the left column, these solutions represent different physical states. For the right column, they represent the same physical state.

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Non-Abelian gauge field, which was introduced in 1954, was initially found not consistent with experimental results.

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1960’s

Breaking of Symmetry

72Yoichiro Nambu (1921- )

73Steven Weinberg (1933- )

74Abdus Salam (1926-1996)

75Sheldon Lee Glashow (1932- )

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Standard Model

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Symmetry Dictates Interaction

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)path(d)action(i

exp

Propagator =

79Richard Feynman (1918-1988)

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The relationship between gauge theory and 20th century mathematics:

Fiber bundlesTopology

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Taken in 1985. From left: Sheldon Chang, S.S. Chern, C.N. Yang.

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Flow

of

Ideas

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The three thematic melodies of the 20th century led to a new understanding of the basic concepts of physics.

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Space

Time

Motion

Energy

Force

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Origin of the three thematic melodies

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Early concepts related to Quantization:

Democritus (~450 bc) Atoms Zeno (~300 bc) Continuity Zhuang-zhou (~300 bc) Continuity

* * * * * Quantization of action (not of matter)

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Early concepts related to Symmetry:

Anaximander (~600 bc)

Pythagoras (~510 bc): Harmony of the Spheres

* * * * *Non-Abelian Lie Groups

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Early concepts related toPhases:

Phases of the MoonCycling of four seasons

* * * * *Flexibility of phases determines equations governing fundamental forces

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Through more than a century of hard work by mathematicians and physicists, these three primordial and inaccurate concepts

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Became the thematic melodies of twentieth century theoretical physics.

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And these thematic melodies are the spirit of today’s theoretical physics.

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They will continue to lead the development of physics in the next thirty to fifty years.