查看“︁玻色弦理論”︁的源代码

{{NoteTA|G1=Physics|G2=Math}}

'''玻色弦理論'''（{{lang-en|Bosonic string theory}}）是最早的[[弦論]]版本，約在1960年代晚期發展。其名稱由來是因為粒子譜中僅含有[[玻色子]]。

1980年代，在弦論的範疇下發現了[[超對稱]]；一個稱作[[超弦理論]]（超對稱弦理論）的新版本弦論成為了研究主題。儘管如此，玻色弦理論仍然是了解[[微擾]]弦理論的有用工具，並且超弦理論中的一些理論困難之處在玻色弦理論中已然現身。

== 疑難 ==
雖然玻色弦理論有許多吸引人的特質，其在成為物理模型理論有兩大缺陷：
# 其只預測[[玻色子]]的存在，然而許多物理粒子為[[費米子]]。
# 其預測了一種具有[[虛數]][[質量]]的弦模式，暗示了此理論在[[迅子凝聚]]過程會有不穩定性。

== 類型 ==

有四種可能的玻色子弦理論，取決於是否允許[[開弦]]以及弦是否具有指定的[[可定向性]]。四種可能理論的光譜示意圖如下：

{| class="wikitable"
|-
! [[玻色弦理论]] || 非正<math>M^2</math>狀態
|-
| 可開弦定向 || [[快子]]、[[引力子]]、[[脹子]]、無質量反對稱張量（massless antisymmetric tensor）
|-
| 可開弦無向 || [[快子]]、[[引力子]]、[[脹子]]
|-
| 閉弦定向 || [[快子]]、[[引力子]]、[[脹子]]、反對稱張量（antisymmetric tensor）、[[U(1)]]、[[矢量玻色子]]
|-
| 閉弦無向 || [[快子]]、[[引力子]]、[[脹子]]
|}

請注意，所有四種理論都有一個負能量快子 (<math>M^2 = - \frac{1}{\alpha'}</math>) 和一個無質量引力子。

== 數學表示 ==

=== 路徑積分表述 ===

玻色子弦理論可以<ref>D'Hoker, Phong</ref>由[[路徑積分]]定義：

: <math> I_0[g,X] = \frac{T}{8\pi} \int_M d^2 \xi \sqrt{g} g^{mn} \partial_m x^\mu \partial_n x^\nu G_{\mu\nu}(x) </math>

{{TransH}}
<math>x^\mu(\xi)</math> is the field on the [[worldsheet]] describing the embedding of the string in 25+1 spacetime; in the Polyakov formulation, <math>g</math> is not to be understood as the induced metric from the embedding, but as an independent dynamical field. <math>G</math> is the metric on the target spacetime, which is usually taken to be the [[Minkowski metric]] in the perturbative theory. Under a [[Wick rotation]], this is brought to a Euclidean metric <math>G_{\mu\nu} = \delta_{\mu\nu}</math>. M is the worldsheet as a [[topological manifold]] parametrized by the <math>\xi</math> coordinates. <math>T</math> is the string tension and related to the Regge slope as <math>T = \frac{1}{2\pi\alpha'}</math>.

<math>I_0</math> has [[Diffeomorphism invariance|diffeomorphism]] and [[Weyl transformation|Weyl invariance]]. Weyl symmetry is broken upon quantization ([[Conformal anomaly]]) and therefore this action has to be supplemented with a counterterm, along with a hypothetical purely topological term, proportional to the [[Euler characteristic]]:

: <math> I = I_0 + \lambda \chi(M) + \mu_0^2 \int_M d^2\xi \sqrt{g} </math>

The explicit breaking of Weyl invariance by the counterterm can be cancelled away in the [[critical dimension]] 26.

Physical quantities are then constructed from the (Euclidean) [[Partition function (quantum field theory)|partition function]] and [[Correlation function (quantum field theory)|N-point function]]:

: <math> Z = \sum_{h=0}^\infty \int \frac{\mathcal{D}g_{mn} \mathcal{D}X^\mu}{\mathcal{N}} \exp ( - I[g,X] ) </math>
: <math> \left\langle V_{i_1} (k^\mu_1) \cdots V_{i_p}(k_p^\mu) \right\rangle = \sum_{h=0}^\infty \int \frac{\mathcal{D}g_{mn} \mathcal{D}X^\mu}{\mathcal{N}} \exp ( - I[g,X] ) V_{i_1} (k_1^\mu) \cdots V_{i_p} (k^\mu_p) </math>

[[File:Sum over genera.png|thumb|right|The perturbative series is expressed as a sum over topologies, indexed by the genus.]]

The discrete sum is a sum over possible topologies, which for euclidean bosonic orientable closed strings are compact orientable [[Riemannian manifold|Riemannian surfaces]] and are thus identified by a genus <math>h</math>. A normalization factor <math>\mathcal{N}</math> is introduced to compensate overcounting from symmetries. While the computation of the partition function corresponds to the [[cosmological constant]], the N-point function, including <math>p</math> vertex operators, describes the scattering amplitude of strings.

The symmetry group of the action actually reduces drastically the integration space to a finite dimensional manifold. The <math>g</math> path-integral in the partition function is ''a priori'' a sum over possible Riemannian structures; however, [[Quotient space (topology)|quotienting]] with respect to Weyl transformations allows us to only consider [[conformal structure]]s, that is, equivalence classes of metrics under the identifications of metrics related by

: <math> g'(\xi) = e^{\sigma(\xi)} g(\xi) </math>

Since the world-sheet is two dimensional, there is a 1-1 correspondence between conformal structures and [[complex manifold|complex structures]]. One still has to quotient away diffeomorphisms. This leaves us with an integration over the space of all possible complex structures modulo diffeomorphisms, which is simply the [[moduli space]] of the given topological surface, and is in fact a finite-dimensional [[complex manifold]]. The fundamental problem of perturbative bosonic strings therefore becomes the parametrization of Moduli space, which is non-trivial for genus <math>h \geq 4</math>.

<!-- The single most important quantity in first quantized bosonic string theory is the N-point scattering amplitude. This treats the incoming and outgoing strings as points, which in string theory are [[tachyon]]s, with momentum ''k''<sub>''i''</sub> which connect to a string world surface at the surface points ''z''<sub>''i''</sub>. It is given by the following [[functional integral]] over all possible embeddings of this 2D surface in 26 dimensions:<ref>Polchinski, Joseph. ''String Theory: Volume I''. Cambridge University Press, p. 173.</ref>

: <math> A_N = \int  D\mu \int  D[X] \exp \left( -\frac{1}{4\pi\alpha} \int \partial_z X_\mu(z,\overline{z}) \partial_{\overline{z}}  X^\mu(z,\overline{z}) \, dz^2 + i \sum_{i=1}^N  k_{i \mu} X^\mu (z_i,\overline{z}_i) \right) </math>

The functional integral can be done because it is a Gaussian to become:

: <math> A_N = \int  D\mu \prod_{0<i<j<N+1} |z_i-z_j|^{2\alpha k_i.k_j} </math>

This is integrated over the various points ''z''<sub>''i''</sub>. Special care must be taken because two parts of this complex region may represent the same point on the 2D surface and you don't want to integrate over them twice. Also you need to make sure you are not integrating multiple times over different parameterizations of the surface. When this is taken into account it can be used to calculate the 4-point scattering amplitude (the 3-point amplitude is simply a delta function):

: <math> A_4 = \frac{ \Gamma (-1+\frac12(k_1+k_2)^2) \Gamma (-1+\frac12(k_2+k_3)^2)  } { \Gamma (-2+\frac12((k_1+k_2)^2+(k_2+k_3)^2)) } </math>

Which is a [[beta function]], known as [[Veneziano amplitude]]. It was this beta function which was apparently found before full string theory was developed. With superstrings the equations contain not only the 10D space-time coordinates X but also the Grassmann coordinates&nbsp;''&theta;''. Since there are various ways this can be done this leads to different string theories.

When integrating over surfaces such as the torus, we end up with equations in terms of [[theta functions]] and elliptic functions such as the [[Dedekind eta function]]. This is smooth everywhere, which it has to be to make physical sense, only when raised to the 24th power. This is the origin of needing 26 dimensions of space-time for bosonic string theory. The extra two dimensions arise as degrees of freedom of the string surface. -->

==== h = 0 ====

At tree-level, corresponding to genus 0, the cosmological constant vanishes: <math> Z_0 = 0 </math>.

The four-point function for the scattering of four tachyons is the Shapiro-Virasoro amplitude:

: <math> A_4 \propto (2\pi)^{26} \delta^{26}(k) \frac{\Gamma(-1-s/2) \Gamma(-1-t/2) \Gamma(-1-u/2)}{\Gamma(2+s/2) \Gamma(2+t/2) \Gamma(2+u/2)} </math>

Where <math>k</math> is the total momentum and <math>s</math>, <math>t</math>, <math>u</math> are the [[Mandelstam variables]].

==== h = 1 ====

[[File:ModularGroup-FundamentalDomain.svg|thumb|right|alt=Fundamental domain for the modular group.| The shaded region is a possible fundamental domain for the modular group.]]Genus 1 is the torus, and corresponds to the [[One-loop Feynman diagram|one-loop level]]. The partition function amounts to:

: <math> Z_1 = \int_{\mathcal{M}_1} \frac{d^2 \tau}{8\pi^2 \tau_2^2} \frac{1}{(4\pi^2 \tau_2)^{12}} \left| \eta(\tau) \right| ^{-48} </math>

<math>\tau</math> is a complex number with positive imaginary part <math>\tau_2</math>; <math>\mathcal{M}_1</math>, holomorphic to the moduli space of the torus, is any [[fundamental domain]] for the [[modular group]] <math>PSL(2,\mathbb{Z})</math> acting on the [[upper half-plane]], for example <math> \left\{ \tau_2 > 0, |\tau|^2 > 1, -\frac{1}{2} < \tau_1 < \frac{1}{2}  \right\} </math>. <math>\eta(\tau)</math> is the [[Dedekind eta function]]. The integrand is of course invariant under the modular group: the measure <math> \frac{d^2 \tau}{\tau_2^2} </math> is simply the [[Poincaré metric]] which has [[SL2(R)|PSL(2,R)]] as isometry group; the rest of the integrand is also invariant by virtue of <math>\tau_2 \rightarrow |c \tau + d|^2 \tau_2 </math> and the fact that <math>\eta(\tau)</math> is a [[modular form]] of weight 1/2.

This integral diverges. This is due to the presence of the tachyon and is related to the instability of the perturbative vacuum.
{{TransF}}

== 相關條目 ==
*[[弦理論]]
*[[快子]]

== 參考 ==
{{reflist}}

=== 參考文獻 ===
{{cite journal
|  title = The geometry of string perturbation theory
|author1=D'Hoker, Eric  |author2=Phong, D. H. |authorlink2=Duong Hong Phong
 |name-list-style=amp |  journal = Rev. Mod. Phys.
|  volume = 60
|  issue = 4
|  pages = 917–1065
|date=Oct 1988

|  publisher = American Physical Society
|bibcode = 1988RvMP...60..917D |doi = 10.1103/RevModPhys.60.917 }}

{{cite journal
 |title           = Complex geometry and the theory of quantum strings
 |author1         = Belavin, A.A.
 |author2         = Knizhnik, V.G.
 |name-list-style = amp
 |journal         = ZhETF
 |volume          = 91
 |issue           = 2
 |pages           = 364–390
 |date            = Feb 1986
 |url             = http://www.jetp.ac.ru/cgi-bin/index/e/64/2/p214?a=list
 |bibcode         = 1986ZhETF..91..364B
 |access-date     = 2022-06-18
 |archive-date    = 2021-02-26
 |archive-url     = https://web.archive.org/web/20210226154142/http://www.jetp.ac.ru/cgi-bin/index/e/64/2/p214?a=list
}}

== 外部鏈接 ==
* [https://web.archive.org/web/20101008035958/http://superstringtheory.com/basics/basic5a.html How many string theories are there?]
* [http://pirsa.org/C09001 PIRSA:C09001 - Introduction to the Bosonic String] {{Wayback|url=http://pirsa.org/C09001 |date=20220616050100 }}

{{String theory}}

[[Category:弦理論]]