Relationship between string theory and quantum field theory
Encyclopedia
Many first principles in quantum field theory
are explained, or get further insight, in string theory
:
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.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
are explained, or get further insight, in string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
:
- Emission and absorption: one of the most basic building blocks of quantum field theory, is the notion that particles (such as electronElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s) can emit and absorb other particles (such as photonPhotonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s). Thus, an electronElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
may just "split" into an electronElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
plus a photonPhotonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
, with a certain probability (which is, roughly, the coupling constantCoupling constantIn physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
). 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. - Coupling constantCoupling constantIn physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
: in quantum field theory this is, roughly, the probability for one particle to emit or absorb another particle, the latter typically being a gauge bosonGauge bosonIn particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...
(a particle carrying a forceFundamental interactionIn particle physics, fundamental interactions are the ways that elementary particles interact with one another...
). In string theory, the coupling constant is no longer a constant, but is rather determined by the abundance of strings in a particular mode, the dilatonDilatonIn particle physics, a dilaton is a hypothetical particle. It also appears in Kaluza-Klein theory's compactifications of extra dimensions when the volume of the compactified dimensions vary....
. Strings in this mode couple to the worldsheetWorldsheetIn string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
curvatureCurvature of Riemannian manifoldsIn mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define it, now known as the curvature tensor...
of other strings, so their abundance through space-time determines the measure by which an average string worldsheetWorldsheetIn string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
will be curved. This determines its probability to split or connect to other strings: the more a worldsheet is curved, it has a higher chance of splitting and reconnecting. - SpinSpin (physics)In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
: each particle in quantum field theory has a particular spin s, which is an internal angular momentumAngular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
. Classically, the particle rotates in a fixed frequency, but this cannot be understood if particles are point-like. In string theory spin is understood by the rotation of the string; For example, a photonPhotonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
with well-defined spin components (i.e. in circular polarizationCircular polarizationIn electrodynamics, circular polarization of an electromagnetic wave is a polarization in which the electric field of the passing wave does not change strength but only changes direction in a rotary type manner....
) looks like a tiny straight line revolving around its center. - Gauge symmetry: in quantum field theory, the mathematical description of physical fields include non-physical states. In order to omit these states from the description of every physical process, a mechanism called gauge symmetry is used. This is true for string theory as well, but in string theory it is often more intuitive to understand why the non-physical states should be disposed of. The simplest example is the photonPhotonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
: a photon is a vector particleElementary particleIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
(it has an inner "arrow" which points to some direction - its polarizationPhoton polarizationPhoton polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photons are completely polarized...
). Mathematically, it can point towards any direction in space-time. Suppose the photon is moving in the z direction; then it may either point towards the x, y or z spatial directions, or towards the t (time) direction (or any diagonal direction). Physically, however, the photon may not point towards the z or t directions (longitudinalLongitudinal waveLongitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...
polarization), but only in the x-y plane (transverseTransverse waveA transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer...
polarization). A 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 worldsheetWorldsheetIn string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
, 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 transverseTransverse waveA transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer...
.
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
Worldsheet
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
under a local change of coordinates and scales.
- renormalizationRenormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
: in particle physicsParticle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
the behaviour of particles in the smallest scales is largely unknown. In order to avoid this difficulty, the particles are treated as point-like objects, and a mathematical tool known as renormalization is used to describe the unknown aspects by only few parameters, which 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. - fermionFermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s: in the bosonic string, a string can be described as an elastic one-dimensional object (i.e. a line) "living" in spacetimeSpacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. 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 fieldField (physics)In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
"living" on the string. This is a fermionic fieldFermionic fieldIn quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi-Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields....
, 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 worldsheetWorldsheetIn string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
) is ultimately responsible for the appearance of fermionFermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s in spacetimeSpacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
: roughly, two strings with arrows drawn on them cannot coexist at the same point in spacetimeSpacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, 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 fermionFermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s in spacetimeSpacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
.