Relationship between string theory and quantum field theory

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Many first principles in quantum field theory are explained, or get further insight, in string theory.

From quantum field theory to string theory

• Emission and absorption: one of the most basic building blocks of quantum field theory, is the notion that particles (such as electrons) can emit and absorb other particles (such as photons). Thus, an electron may just "split" into an electron plus a photon, with a certain probability (which is roughly the coupling constant). This is described in string theory as one string splitting into two. This process is an integral part of the theory. The mode on the original string also "splits" between its two parts, resulting in two strings which possibly have different modes, representing two different particles.
• space-time determines the measure by which an average string worldsheet
  will be curved. This determines its probability to split or connect to other strings: the more a worldsheet is curved, the higher a chance of splitting and reconnecting it has.
• components (i.e. in circular polarization
  ) looks like a tiny straight line revolving around its center.
• gauge symmetry is used to dispose of the non-physical states. In string theory, a photon is described by a tiny oscillating line, with the axis of the line being the direction of the polarization (i.e. the inner direction of the photon is the axis of the string which the photon is made of). If we look at the worldsheet, the photon will look like a long strip which stretches along the time direction with an angle towards the z-direction (because it is moving along the z-direction as time goes by); its short dimension is therefore in the x-y plane. The short dimension of this strip is precisely the direction of the photon (its polarization) in a certain moment in time. Thus the photon cannot point towards the z or t directions, and its polarization must be transverse
  .

Note: formally, gauge symmetries in string theory are (at least in most cases) a result of the existence of a global symmetry together with the profound

gauge symmetry of string theory, which is the symmetry of the worldsheet

under a local change of coordinates and scales.

• Renormalization: in particle physics the behaviour of particles in the smallest scales is largely unknown. In order to avoid this difficulty, the particles are treated as fields behaving according to an "effective field theory" at low energy scales, and a mathematical tool known as renormalization is used to describe the unknown aspects of this effective theory using only a few parameters. These parameters can be adjusted so that calculations give adequate results. In string theory, this is unnecessary since the behaviour of the strings is presumed to be known to every scale.
• Fermions: in the bosonic string, a string can be described as an elastic one-dimensional object (i.e. a line) "living" in spacetime. In superstring theory, every point of the string is not only located at some point in spacetime, but it may also have a small arrow "drawn" on it, pointing at some direction in spacetime. These arrows are described by a field "living" on the string. This is a fermionic field, because at each point of the string there is only one arrow; thus one cannot bring two arrows to the same point. This fermionic field (which is a field on the worldsheet) is ultimately responsible for the appearance of fermions in spacetime: roughly, two strings with arrows drawn on them cannot coexist at the same point in spacetime, because then one would effectively have one string with two sets of arrows at the same point, which is not allowed, as explained above. Therefore two such strings are fermions in spacetime.^[1]

Notes