Ann. Henri Poincaré 4, Suppl. 1 (2003) S9 – S14
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/01S9-6
DOI 10.1007/s00023-003-0902-5
Annales Henri Poincaré
The Thematic Melodies of Twentieth Century
Theoretical Physics∗
Chen Ning Yang
1 Introduction
Among the main historic achievements brought forth by physics in the 20th century:
• Man discovered, for the first time since our ancestors discovered fire, the
second and vastly stronger source of energy: nuclear power.
• Man learned to manipulate electrons to create the transistor which led to the
modern computer, thereby increasing human productivity greatly.
• Man learned how to probe into structures of atomic dimensions, thereby
discovered the double-helix, leading to the birth of biotechnology.
• Man took first steps on the moon.
However, from the viewpoint of physicists, the most important advances are the
profound revolutions in our understanding of the basic concepts of physics: Space,
Time, Motion, Energy and Force.
There are three main strands that had persistently woven through all conceptual advances in physics in the 20th century, three Thematic Melodies in Symphonic Music. They are Quantization, Symmetry and Phase Factor.
2 Quantization
Main dates are 1900: Planck, 1905: Einstein, 1913: Bohr.
“It was the spring of hope, it was the winter of despair”
At present I am myself most optimistic as regards the future of the theory.
Bohr to Rutherford 1918.
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.
Heisenberg’s mechanics has restored my zest for life. Pauli to Kronig, October
9, 1925.
∗ This lecture is derived from the PowerPoint lecture of Professor Yang, respecting the exact
original style as much as possible. The many accompanying photographs were not included for
lack of space.
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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 indefatigable. We would just be crushed in that juggernaut.
Kramers to Klein 1927, quoted in Pais “Genius of Science”, p.159 (2000).
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 hurried conferences, of debate, criticism, and brilliant mathematical improvisation. For those who participated, it was a time of
creation; there was terror as well as exaltation in their new insight. It will probably not 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 it is unlikely to be known to
any poet or any historian. J.R. Oppenheimer, Reith Lectures 1953.
The actors characteristics were
• Pauli: Power
• Fermi: Solidity, Strength
• Heisenberg: Deep Insigh
• Dirac: Cartesian Purity
3 Symmetry (= invariance)
Main dates are 1905: Einstein, 1908: Minkowski.
Superfluous learnedness?
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 the coordinates in
the four-dimensional continuum. This happened in 1908. Einstein, Autobiographical Notes, in “Albert Einstein”, ed. P.A. Schilpp, p.67.
With the introduction of quantum mechanics in 1925, symmetry became very
important. The mathematical language for symmetry is groups.
It has been rumored that the group pest is gradually being cut out of quantum
physics. H. Weyl, Nov. 1930.
Symmetry gradually became a thematic melody (1927–1970) of atomic,
molecular physics, nuclear physics and elementary particle physics. A great shock
was created by Prof. C. S. Wu in 1957: Parity Nonconservation in Weak Interactions.
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.
Never before or afterward have I seen him so excited about physics. Heisenberg
1978.
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The Thematic Melodies of Twentieth Century Theoretical Physics
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Figure 1: The five regular solids with maximum symmetry. Reprinted from A.V.
Shubnikov and V.A. Koptsik, Symmetry in Science and Art (Plenum, 1974).
4 Phase Factor
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.
The phase factor is eiθ , θ = 0 to 360. The phase factor became important
through the proposal of Weyl in 1918, modified in 1927. Weyl introduced
in 1918
√
the gauge factor eθ Then London and Fock added in 1927 i = −1, so that the
gauge factor eθ became the phase factor eiθ .
Gauge Theory follows from the flexibility of the phase factor. It leads to the
electromagnetic equation
The Phase Factor of Weyl in 1918, exp[−(e/γ)Aµ dxµ ], is a stretch factor. In
1922 Schrödinger remarks that along a Bohr orbit, the exponent is −nh/γ with n
integer, a “Remarkable Property”. Since γ = −iη this stretch factor is 1.
The de Broglie interpretation of the quantum rules seems to me to be related in some ways to my note in the Zeitschrift Für. Phys. 12, 13, 1922,. . . The
mathematical situation is, as far as I can see, the same, only from me much more
formal, less elegant and not really shown generally. Schrödinger to Einstein, Nov
3, 1925.
The three thematic melodies were introduced in the first half of the century,
their developments in the next half century were Developments, Variations and
Intertwinings.
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Figure 2: History of the Phase Factor
The generalization of Gauge Symmetry is from p−eA to p−eB. It is motivated
by the discovery of more and more strange particles which need a general principle
for interaction.
Conservation of charge => electromagnetic field
Conservation of energy => gravitational field
Why other conservation laws do not lead to specific field?
Conservation laws were related to global gauge transformation. It is not consistent with the localized field concept.
Non-Abelian Gauge Field uses the Mathematical Language of Symmetry,
namely Groups, created by Galois (1811–1832) and Lie (1842–1899). The simplest
Lie Group is the Phase Factor eiθ . Non-Abelian Lie groups are generalizations of
this Phase Factor.
Flexibility in Definition of Phase lead in 1929 to the understanding that
ElectroMagnetism is Gauge Theory.
Flexibility Generalized lead in 1954 to Non-Abelian Gauge Theory. NonAbelian gauge field, which was introduced in 1954, was initially found not consistent with experimental results.
1960s: Breaking of Symmetry. Standard Model Symmetry Dictates Interaction.
The propagator is exp ηi (action) d(path) (Feynman).
The relationship between gauge theory and 20th century mathematics lead
to consider Fiber bundles and Topology.
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The Thematic Melodies of Twentieth Century Theoretical Physics
Figure 3: The flow of Ideas
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5 Conclusion
The three thematic melodies of the 20th century led to a new understanding of
the basic concepts of physics, Space, Time, Motion, Energy, Force.
One can trace back the origin of the three thematic melodies:
Early concepts related to Quantization:
• Democritus (450 bc), Atoms
• Zeno (300 bc), Continuity
• Zhuang-zhou (300 bc), Continuity
=> Quantization of action (not of matter)
Early concepts related to Symmetry:
• Anaximander (600 bc)
• Pythagoras (510 bc), Harmony of the Spheres
=> Non-Abelian Lie Groups
Early concepts related to Phases:
• Phases of the Moon
• Cycling of four seasons
=> Flexibility of phases determines equations governing fundamental forces
Through more than a century of hard work by mathematicians and physicists,
these three primordial and inaccurate concepts became the thematic melodies
of twentieth century theoretical physics. And these thematic melodies are the
underlying spirit of todays theoretical physics. They will continue to lead the
development of physics in the next thirty to fifty years.
Chen N. Yang
Room 352A 3rd Floor
Science Centre North Block
The Chinese University of Hong Kong
Shatin, Hong Kong
China
Email: [email protected]