View source for Physics:Free field

{{Short description|Physical field theory with no forces/interactions}}
{{Context|see discussion|date=October 2009}}-->
{{Quantum field theory}}
In [[Physics:Physics|physics]] a '''free field''' is a [[Physics:Field|field]] '''without [[Physics:Fundamental interaction|interaction]]s''', which is described by the terms of motion and mass.

== Description ==

In [[Physics:Classical physics|classical physics]], a '''free field''' is a field whose [[Physics:Equations of motion|equations of motion]] are given by linear [[Partial differential equation|partial differential equation]]s. Such linear PDE's have a unique solution for a given initial condition.

In [[Physics:Quantum field theory|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 [[Physics:Lagrangian (field theory)|Lagrangian]]. We can differentiate distributions by defining their derivatives via differentiated test functions. 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|commutator]] (for [[Physics:Boson|boson]]s)/anticommutator (for fermions) of two smeared fields is i times the [[Physics:Peierls bracket|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 [[Physics:Minkowski space|Minkowski space]], we may choose the unitary irrep containing a [[Physics:Vacuum state|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=0</math>.

This is a bosonic field. Let's call the distribution given by the [[Physics:Peierls bracket|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 [[Physics:Canonical commutation relation|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 ==
*[[Physics:Normal order|Normal order]]
*[[Physics:Wick's theorem|Wick's theorem]]

== References ==
* Michael E. Peskin and Daniel V. Schroeder, ''An Introduction to Quantum Field Theory'', Addison-Wesley, Reading, 1995. p19-p29

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[[Category:Quantum field theory]]

{{Sourceattribution|Free field}}