Pei-Ming HoDepartment of Physics,

NTUMarch 2006

Spacetime for Strings

What is spacetime?

Spacetime in Einstein’s theory

• The Riemannian geometric property (notion of distance and curvature, etc.) of the spacetime is dynamically determined by the theory.

• Gravity geometrized.

• Field equations determine the motion of particles.

Why is D = 3+1 ?

Einstein’s theory vs. Strings

• The dimension of spacetime, the geometric structure (Riemann) are put in by hand.

• Singularities.• Quantization? information paradox• String theory is a consistent (and comput

able) theory of quantum gravity approximated by Einstein’s theory at low energies.

at Planck scale

A naïve combination of GR and QM suggests that the classical notion of spacetime breaks down at the Planck scale (1019 GeV, 10-33 cm).

m

What is quantum spacetime?We need a theory of quantum gravity.String theory is essentially our only hope.

What is string

theory?

particle → string

oscillation modes → particles-1-0.500.51

-1-0.500.51

0

10

20

30

unique interaction

solitons

• There are solitons called D-branes.

• D-branes come in various dimensions.

• There are open strings ending on D-branes.

This idea about strings leads to very complicated mathematical problems, but surprisingly the theory has many nice and interesting properties.

“弦論” 符合網頁 (2001)

1. UFO:

•飛碟到底從哪裡來？•…克氏理論」有五維的時空，以及「超弦理論」有十維的時空。目前科學家矚目的「超弦理論 (superstring theory) 」，是根據「超對稱 (supersymmetry) 」和「弦論 (string theory) 」兩個理論結合的一種最新的物理理論。

•外星人與飛碟真相的研究。•從幽浮現象探討宇宙三度空間 ∙

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•霍金預言百年後人類將殖民外星… ∙

3. Not related to string theory YET:•撥動陽光之弦－論槐華詩歌的美學追求…•吳弦，論指揮管制戰的真義… •『紀弦論現代詩』…•前者以趺陽脈弦論病，這條是以寸口脈弦論病… ∙

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•史蓋皮諾立刻向貝尼特說出他目前的研究計畫，那是叫做「弦論」的玩意，處理次原子世界的現象 ∙

5. 學術界：•李仁吉教授，美國賓州大學物理博士，專長：場論、弦論。

• http://www.phys.ntu.edu.tw/~pmho

台大弦論小組的網站 ∙

“弦論” 符合網頁 (2006)約有 34,000 項符合弦論的查詢結果

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2. 學術界…3. Not related to string theory YET:二胡專輯《胡弦論雨》 七弦論考 ∙

Strings everywhere!

about string theory

• String theory is not yet experimentally verified. (But this is not a geometry-independent question…)

• String Theory includes Quantum Gravity.• The spacetime is 10 dimensional for super

string theory (11 for M theory, 12 for F theory, 5 fo

r IIB in AdS, 1 for BFSS and 0 for IKKT).

Kaluza-Klein w. Calabi-Yau

Brane WorldBrane World

fundamental strings

• Spacetime = target space of the fields living on the string worldsheet:

• Dimension is determined by symmetry.

• High energy limit is smooth, not spiky.

• Change of variables / field redefinition different interpretation of spacetime

1,,1,0),,( DX

T-duality

Winding modes are interchanged with momentum modes.

probe-dependence

• fundamental string or soliton (D-brane)« S-duality interchanges F1 and D1. »

• Geometric properties can not be completely separated from interactions.Example:Noncommutative geometry[ x, y ] = ic Δx Δy ≥ c

theory-dependence

• M theory (11D) IIA superstring theory (10D)

• All 5 superstring theories are equivalent.

• Holography:

Q. Gravity in higher dimensions Local field theories in lower dimensions.

Example: AdS/CFT duality.

IIB superstring in AdS SYM in 4 dim.

道可道非常道 名可名非常名