Quantum topology is a branch of mathematics that connects
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
with
low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot th ...
.
Dirac notation
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associated with
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s and the related embedding of one space within another such as
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
s and
links in three-dimensional space. This
bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
of kets and bras can be generalised, becoming maps of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s associated with
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s that allow
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
s.
Topological
entanglement involving
linking and
braiding
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
can be intuitively related to
quantum entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
.
See also
*
Topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathem ...
*
Reshetikhin–Turaev invariant
In the mathematical field of quantum topology, the Reshetikhin–Turaev invariants (RT-invariants) are a family of quantum invariants of framed links.
Such invariants of framed links also give rise to invariants of 3-manifolds via the Dehn surgery ...
References
External links
Quantum Topology a journal published by EMS Publishing House
Quantum mechanics
Topology
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