Hankel transform: Difference between revisions

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en>Anrnusna
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Added a section on an alternative definition and its domain of definition. Moved a couple sentences.
 
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{{Context|see discussion|date=October 2009}}-->
{{Quantum field theory}}
In [[physics]] a '''free field''' is a [[Field (physics)|field]] '''without [[fundamental interaction|interaction]]s''', which is described by the terms of motion and mass.
 
== Description ==
 
In [[classical physics]], a '''free field''' is a field whose [[equations of motion]] are given by [[linear]] [[partial differential equation]]s. Such linear PDE's have a unique solution for a given initial condition.
 
In [[quantum field theory]], an [[operator valued distribution]] is a '''free field''' if it satisfies some linear partial differential equations such that the corresponding case of the same linear PDEs for a classical field (i.e. not an operator) would be the [[Euler-Lagrange equation]] for some [[quadratic polynomial|quadratic]] [[Lagrangian]]. We can differentiate distributions by defining their derivatives via differentiated [[test function]]s. See [[Schwartz distribution]] for more details. Since we are dealing not with ordinary distributions but operator valued distributions, it is understood these PDEs aren't constraints on states but instead a description of the relations among the smeared fields. Beside the PDEs, the operators also satisfy another relation, the commutation/anticommutation relations.
 
== Canonical Commutation Relation ==
Basically, [[commutator]] (for [[boson]]s)/[[anticommutator]] (for [[fermions]]) of two smeared fields is i times the [[Peierls bracket]] of the field with itself (which is really a distribution, not a function) for the PDEs smeared over both test functions. This has the form of a [[CCR/CAR algebra]].
 
CCR/CAR algebras with infinitely many degrees of freedom have many inequivalent irreducible unitary representations. If the theory is defined over [[Minkowski space]], we may choose the unitary irrep containing a [[vacuum state]] although that isn't always necessary.
 
==Example==
Let φ be an operator valued distribution and the (Klein-Gordon) PDE be
 
:<math>\partial^\mu \partial_\mu \phi+m^2 \phi^2=0</math>.
 
This is a bosonic field. Let's call the distribution given by the Peierls bracket Δ.
 
Then,
 
:<math>\{\phi(x),\phi(y)\}=\Delta(x;y)</math>
 
where here, φ is a classical field and {,} is the Peierls bracket.
   
Then, the [[canonical commutation relation]] is
 
:<math>[\phi[f],\phi[g]]=i\Delta[f,g] \,</math>.
 
Note that Δ is a distribution over two arguments, and so, can be smeared as well.
 
Equivalently, we could have insisted that
 
:<math>\mathcal{T}\{[((\partial^\mu \partial_\mu+m^2)\phi)[f],\phi[g]]\}=-i\int d^dx f(x)g(x)</math>
 
where <math>\mathcal{T}</math> is the [[time ordering]] operator and that if the supports of f and g are spacelike separated,
 
:<math>[\phi[f],\phi[g]]=0</math>.
 
== See also ==
*[[Normal order]]
*[[Wick's theorem]]
 
== References ==
* Michael E. Peskin and Daniel V. Schroeder, ''An Introduction to Quantum Field Theory'', Addison-Wesley, Reading, 1995. p19-p29
 
{{DEFAULTSORT:Free Field}}
[[Category:Quantum field theory]]

Latest revision as of 20:44, 28 July 2014



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