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Idea

In a sigma-model quantum field theory , afield history is a morphism $\phi \colon \Sigma \to X$ for an$\mathrm{<span><del\; class="diffmod">\; \Sigma </del><ins\; class="diffmod">\; n</ins></span>}$ \Sigma n - an$n$-dimensional manifold or similar. One is to think of this as being thetrajectory: of an $(n-1)$-brane propagating in the target space $X$.$\Sigma$ or similar. One is to think of this as the trajectory of an $(n-1)$-brane propagating in the target space $X$.

For Cases the of case$n=1$ n = 1 : (for instance therelativistic particle, a 0-brane) the term worldline for $\Sigma$ has a long tradition. Accordingly one calls $\Sigma$ the worldvolume of the given $(n-1)$-brane when $n \gt 1$. For the case $n=2$ (the case of relevance in string theory) one also says worldsheet.

Hence generally for any field theory defined on a worldvolume or spacetime $\Sigma$, and with type of fields determined by a field bundle $E \overset{fb}{\to} \Sigma$, one may think of a section of the field bundle as a field trajectory.

1. $n = 1$ (for instance, the relativistic particle, a 0-brane): $\Sigma$ is called the “worldline”
2. $n \gt 1$: $\Sigma$ is called the “worldvolume”
3. $n = 2$ (relevant in string theory): $\Sigma$“ is called the ”worldsheet“

The Hence, generally for any space field theory of defined all on these a is thespace of trajectoriesworldvolume (space or of histories).spacetime $\Sigma$, and with type of fields determined by a field bundle $E \overset{fb}{\to} \Sigma$, one may think of a section of the field bundle as a field trajectory.

The space of all field histories is the space of trajectories (space of field histories).

Related concepts

• worldline formalism (for scattering amplitudes in QFT)

• sigma-model

  • worldvolume

    • worldsheet

      • Riemann surface, elliptic curve, algebraic curve

      • worldsheet parity operator

  • target space

  • background gauge field

  • genus

• worldsheet instanton

• world sheets for world sheets