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(which is defined as the integrand in Eq.(1)). |

The spectrum of states is calculated from the Hamiltonian: | ---------- (16) |

For a closed string it takes the form: | ---------- (17) |

For an open string: | ---------- (18) |

---------- (19) |

Each term in the sum in Eqs.(17) and (18) is similar to the number operator in quantum field theory. Eq.(15) shows that the norm for the time-like component is negative, i.e.,

---------- (21) |

X

In this form, X

---------- (22b) |

---------- (22c) |

---------- (22d) |

It was found in the 1960s that the spin (or angular momentum) of a family of resonances (short-lived elementary particles) is related to the square of mass by a simple line on a graph, which is known as Regge trajectory. Figures 03a and 03b show the theoretical derivation of such relationship for a few low-lying string states of the closed and open strings respectively. | ||

## Figure 03a Regge Trajectories, closed string [view large image] |
## Figure 03b Regge Trajectories, open string [view large image] |

Considering the

Figures 03a and 03b.

String Type | String State | Spin | M^{2} |
Name |
---|---|---|---|---|

Open | 0 | -2 | vacuum with tachyon | |

Open | 0 | 0 | massless scalar | |

Open | 0 | 2 | massive scalar | |

Open | 1 | 0 | massless vector (photon) | |

Open | 1 | 2 | massive vector | |

Open | 2 | 2 | massive spine-2 | |

Closed | 0 | -8 | vacuum with tachyon | |

Closed | 0 | 0 | massless scalar | |

Closed | 2 | 0 | massless spin-2 (graviton) |

Eqs. (19) and (22c,d) in the quantized theory of string show that each vibrational mode corresponds to a particle with distinct spin and mass. Unfortunately, as shown in Table 01 only the massless states have counterparts of particles in the real world (observed in the case of photon or not-yet-observed in the case of graviton), other states are either unreal (such as the tachyon) or too heavy in the range of 10

See an "Introduction to Superstring Theory".

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