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In [[mathematics]], the '''smash product''' of two [[pointed space]]s (i.e. [[topological space]]s with distinguished basepoints) ''X'' and ''Y'' is the [[quotient space|quotient]] of the [[product space]] ''X'' &times; ''Y'' under the identifications (''x'',&nbsp;''y''<sub>0</sub>)&nbsp;∼&nbsp;(''x''<sub>0</sub>,&nbsp;''y'') for all ''x''&nbsp;∈&nbsp;''X'' and ''y''&nbsp;∈&nbsp;''Y''. The smash product is usually denoted ''X''&nbsp;∧&nbsp;''Y'' or ''X''&nbsp;⨳&nbsp;''Y''. The smash product depends on the choice of basepoints (unless both ''X'' and ''Y'' are [[homogeneous space|homogeneous]]).


In [[physics]], a '''bound state''' describes a system where a [[particle]] is subject to a [[Potential Energy|potential]] such that the particle has a tendency to remain localised in one or more regions of space. The potential may be either an external potential, or may be the result of the presence of another particle.
One can think of ''X'' and ''Y'' as sitting inside ''X'' &times; ''Y'' as the [[subspace (topology)|subspaces]] ''X'' &times; {''y''<sub>0</sub>} and {''x''<sub>0</sub>} &times; ''Y''. These subspaces intersect at a single point: (''x''<sub>0</sub>, ''y''<sub>0</sub>), the basepoint of ''X'' &times; ''Y''. So the union of these subspaces can be identified with the [[wedge sum]] ''X'' ∨ ''Y''. The smash product is then the quotient
:<math>X \wedge Y = (X \times Y) / (X \vee Y). \, </math>


In [[quantum mechanics]] (where the number of particles is conserved), a bound state is a state in [[Hilbert space]] that corresponds to two or more particles whose [[interaction energy]] is less than the total energy of each separate particle, and therefore these particles cannot be separated unless [[energy]] is spent. The [[energy spectrum]] of a bound state is discrete, unlike the continuous spectrum of isolated particles. (Actually, it is possible to have unstable bound states with a positive interaction energy provided that there is an "energy barrier" that has to be [[quantum tunnelling|tunnelled]] through in order to decay. This is true for some [[Radionuclide|radioactive nuclei]] and for some [[electret]] materials able to carry electric charge for rather long periods.)
The smash product has important applications in [[homotopy theory]], a branch of [[algebraic topology]]. In homotopy theory, one often works with a different [[category (mathematics)|category]] of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two [[CW complex]]es is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories.
 
In general, a stable bound state is said to exist in a given potential of some dimension if stationary wavefunctions exist (normalized in the range of the potential). The energies of these wavefunctions are negative.
 
In [[theory of relativity|relativistic]] [[quantum field theory]], a stable bound state of n particles with masses m<sub>1</sub>, ..., m<sub>n</sub> shows up as a [[pole (complex analysis)|pole]] in the [[S-matrix]] with a center of mass energy which is less than m<sub>1</sub>+...+m<sub>n</sub>. An [[unstable]] bound state (see [[resonance]]) shows up as a pole with a [[complex number|complex]] center of mass energy.


==Examples==
==Examples==
[[Image:Particle overview.svg|thumb|400px|An overview of the various families of elementary and composite particles, and the theories describing their interactions]]
*The smash product of any pointed space ''X'' with a [[0-sphere]] is homeomorphic to ''X''.
* A [[proton]] and an [[electron]] can move separately; the total center-of-mass energy is positive, and such a pair of particles can be described as an ionized atom. Once the electron starts to "orbit" the proton, the energy becomes negative, and a bound state – namely the [[hydrogen atom]] – is formed. Only the lowest energy bound state, the [[ground state]] is stable. The other [[excited state]]s are unstable and will decay into bound states with less energy by emitting a [[photon]].
*The smash product of two circles is a quotient of the [[torus]] homeomorphic to the 2-sphere.
* A [[Atomic nucleus|nucleus]] is a bound state of [[proton]]s and [[neutron]]s ([[nucleon]]s).
*More generally, the smash product of two spheres ''S''<sup>''m''</sup> and ''S''<sup>''n''</sup> is [[homeomorphic]] to the sphere ''S''<sup>''m''+''n''</sup>.
* A [[positronium]] "atom" is an [[resonance|unstable bound state]] of an [[electron]] and a [[positron]]. It decays into [[photon]]s.
*The smash product of a space ''X'' with a circle is homeomorphic to the [[reduced suspension]] of ''X'':
* The [[proton]] itself is a bound state of three [[quark]]s (two [[up quark|up]] and one [[down quark|down]]; one [[Quantum chromodynamics|red]], one [[Quantum chromodynamics|green]] and one [[Quantum chromodynamics|blue]]). However, unlike the case of the hydrogen atom, the individual quarks can never be isolated. See [[color confinement|confinement]].
*:<math> \Sigma X \cong X \wedge S^1. \, </math>
 
*The ''k''-fold iterated reduced suspension of ''X'' is homeomorphic to the smash product of ''X'' and a ''k''-sphere
==In mathematical quantum physics==
*:<math> \Sigma^k X \cong X \wedge S^k. \, </math>
Let <math>H</math> be a complex separable Hilbert space, <math> U = \lbrace U(t) \mid t \in \mathbb{R} \rbrace </math> be a one-parametric group of unitary operators on <math> H </math> and <math>\rho = \rho(t_0) </math> be a statistical operator on <math>H</math>. Let <math>A</math> be an observable on <math>H</math> and let <math>\mu(A,\rho)</math> be the induced probability distribution of <math>A</math> with respect to <math>\rho</math> on the Borel <math>\sigma</math>-algebra on <math>\mathbb{R}</math>. Then the evolution of <math>\rho</math> induced by <math>U</math> is said to be '''bound''' with respect to <math>A</math> if <math>\lim_{R \rightarrow \infty} \sum_{t \geq t_0} \mu(A,\rho(t))(\mathbb{R}_{> R}) = 0 </math>, where <math>\mathbb{R}_{>R} = \lbrace x \in \mathbb{R} \mid x > R \rbrace </math>.
* In [[domain theory]], taking the product of two domains (so that the product is strict on its arguments).
 
'''Example:'''
Let <math>H = L^2(\mathbb{R}) </math> and let <math>A</math> be the position observable. Let <math>\rho = \rho(0) \in H</math> have compact support and <math>[-1,1] \subseteq \mathrm{Supp}(\rho)</math>.
 
* If the state evolution of <math>\rho</math> "moves this wave package constantly to the right", e.g. if <math>[t-1,t+1] \in \mathrm{Supp}(\rho(t)) </math> for all <math>t \geq 0</math>, then <math>\rho</math> is not a bound state with respect to the position.


* If <math>\rho</math> does not change in time, i.e. <math>\rho(t) = \rho</math> for all <math>t \geq 0</math>, then <math>\rho</math> is a bound state with respect to position.
==As a symmetric monoidal product==
For any pointed spaces ''X'', ''Y'', and ''Z'' in an appropriate "convenient" category (e.g. that of [[compactly generated space]]s) there are natural (basepoint preserving) [[homeomorphism]]s
:<math>\begin{align}
X \wedge Y &\cong Y\wedge X, \\
(X\wedge Y)\wedge Z &\cong X \wedge (Y\wedge Z).
\end{align}</math>
However, for the naive category of pointed spaces, this fails. See the following discussion on MathOverflow.<ref>Omar Antolín-Camarena (mathoverflow.net/users/644), In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?, http://mathoverflow.net/questions/76594 (version: 2011-09-28)</ref>


* More generally: If the state evolution of <math>\rho</math> "just moves <math>\rho</math> inside a bounded domain", then <math>\rho</math> is also a bound state with respect to position.
These isomorphisms make the appropriate [[category of pointed spaces]] into a [[symmetric monoidal category]] with the smash product as the monoidal product and the pointed [[0-sphere]] (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of [[tensor product]] in an appropriate category of pointed spaces.


==See also==
==Adjoint relationship==
*[[Composite field]]
[[Adjoint functors]] make the analogy between the tensor product and the smash product more precise. In the category of [[module (mathematics)|''R''-modules]] over a [[commutative ring]] ''R'', the tensor functor (&ndash; ⊗<sub>''R''</sub> ''A'') is left adjoint to the internal [[Hom functor]] Hom(''A'',&ndash;) so that:
*[[Resonance]]
:<math>\mathrm{Hom}(X\otimes A,Y) \cong \mathrm{Hom}(X,\mathrm{Hom}(A,Y)).</math>
*[[Bethe-Salpeter equation]]
In the [[category of pointed spaces]], the smash product plays the role of the tensor product. In particular, if ''A'' is [[locally compact Hausdorff]] then we have an adjunction
:<math>\mathrm{Hom}(X\wedge A,Y) \cong \mathrm{Hom}(X,\mathrm{Hom}(A,Y))</math>
where Hom(''A'',''Y'') is the space of based continuous maps together with the [[compact-open topology]].


{{Particles}}
In particular, taking ''A'' to be the [[unit circle]] ''S''<sup>1</sup>, we see that the suspension functor Σ is left adjoint to the [[loop space]] functor Ω.
{{Chemical bonds}}
:<math>\mathrm{Hom}(\Sigma X,Y) \cong \mathrm{Hom}(X,\Omega Y).</math>


{{DEFAULTSORT:Bound State}}
==References==
[[Category:Quantum mechanics]]
{{Reflist}}
[[Category:Quantum field theory]]
*{{Hatcher AT}}


[[ca:Partícula composta]]
{{DEFAULTSORT:Smash Product}}
[[cy:Cyflwr rhwym]]
[[Category:Topology]]
[[de:Gebundener Zustand]]
[[Category:Homotopy theory]]
[[et:Liitosakesed]]
[[Category:Binary operations]]
[[es:Partícula compuesta]]
[[fr:État lié]]
[[ja:束縛状態]]
[[pt:Partícula composta]]
[[ur:حالت پیوند]]

Revision as of 16:13, 12 August 2014

Template:One source In mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) X and Y is the quotient of the product space X × Y under the identifications (x, y0) ∼ (x0, y) for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y or X ⨳ Y. The smash product depends on the choice of basepoints (unless both X and Y are homogeneous).

One can think of X and Y as sitting inside X × Y as the subspaces X × {y0} and {x0} × Y. These subspaces intersect at a single point: (x0, y0), the basepoint of X × Y. So the union of these subspaces can be identified with the wedge sum X ∨ Y. The smash product is then the quotient

X∧Y=(X×Y)/(X∨Y).

The smash product has important applications in homotopy theory, a branch of algebraic topology. In homotopy theory, one often works with a different category of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two CW complexes is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories.

Examples

  • The smash product of any pointed space X with a 0-sphere is homeomorphic to X.
  • The smash product of two circles is a quotient of the torus homeomorphic to the 2-sphere.
  • More generally, the smash product of two spheres Sm and Sn is homeomorphic to the sphere Sm+n.
  • The smash product of a space X with a circle is homeomorphic to the reduced suspension of X:
    ΣX≅X∧S1.
  • The k-fold iterated reduced suspension of X is homeomorphic to the smash product of X and a k-sphere
    ΣkX≅X∧Sk.
  • In domain theory, taking the product of two domains (so that the product is strict on its arguments).

As a symmetric monoidal product

For any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g. that of compactly generated spaces) there are natural (basepoint preserving) homeomorphisms

X∧Y≅Y∧X,(X∧Y)∧Z≅X∧(Y∧Z).

However, for the naive category of pointed spaces, this fails. See the following discussion on MathOverflow.[1]

These isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of tensor product in an appropriate category of pointed spaces.

Adjoint relationship

Adjoint functors make the analogy between the tensor product and the smash product more precise. In the category of R-modules over a commutative ring R, the tensor functor (– ⊗R A) is left adjoint to the internal Hom functor Hom(A,–) so that:

Hom(X⊗A,Y)≅Hom(X,Hom(A,Y)).

In the category of pointed spaces, the smash product plays the role of the tensor product. In particular, if A is locally compact Hausdorff then we have an adjunction

Hom(X∧A,Y)≅Hom(X,Hom(A,Y))

where Hom(A,Y) is the space of based continuous maps together with the compact-open topology.

In particular, taking A to be the unit circle S1, we see that the suspension functor Σ is left adjoint to the loop space functor Ω.

Hom(ΣX,Y)≅Hom(X,ΩY).

References

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  1. ↑ Omar Antolín-Camarena (mathoverflow.net/users/644), In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?, http://mathoverflow.net/questions/76594 (version: 2011-09-28)
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