Strings

Strings

The motion of a point-like particle can be described by drawing a graph of its position with respect to time. The resulting picture depicts the worldline of the particle inspacetime. In an analogous way, one can draw a graph depicting the progress of a string as time passes. The string, which looks like a small line by itself, will sweep out a two-dimensional surface known as the worldsheet. The different string modes (giving rise to different particles, such as the photon or graviton) appear as waves on this surface.

closed string looks like a small loop, so its worldsheet will look like a pipe. An open string looks like a segment with two endpoints, so its worldsheet will look like a strip. In a more mathematical language, these are both Riemann surfaces, the strip having a boundary and the pipe none.

Interaction in the subatomic world: world lines of point-like particles in the Standard Model or a world sheet swept up by closed strings in string theory

Strings can join and split. This is reflected by the form of their worldsheet, or more precisely, by its topology. For example, if a closed string splits, its worldsheet will look like a single pipe splitting into two pipes. This topology is often referred to as a pair of pants (see drawing at right). If a closed string splits and its two parts later reconnect, its worldsheet will look like a single pipe splitting to two and then reconnecting, which also looks like a torus connected to two pipes (one representing the incoming string, and the other representing the outgoing one). An open string doing the same thing will have a worldsheet that looks like anannulus connected to two strips.

In quantum mechanics, one computes the probability for a point particle to propagate from one point to another by summing certain quantities called probability amplitudes. Each amplitude is associated with a different worldline of the particle. This process of summing amplitudes over all possible worldlines is called path integration. In string theory, one computes probabilities in a similar way, by summing quantities associated with the worldsheets joining an initial string configuration to a final configuration. It is in this sense that string theory extends quantum field theory, replacing point particles by strings. As in quantum field theory, the classical behavior of fields is determined by an action functional, which in string theory can be either the Nambu–Goto action or the Polyakov action.

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