Free field

Free field

Classically, a free field has equations of motion given by linear partial differential equations. 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 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.

Basically, commutator (for bosons)/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 PDE's 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

:partial^mu partial_mu phi+m^2 phi^2=0.

This is a bosonic field. Let's call the distribution given by the Peierls bracket Δ.

Then,

:{phi(x),phi(y)}=Delta(x;y)

where here, φ is a classical field and {,} is the Peierls bracket. Then, the canonical commutation relation relation is

: [phi [f] ,phi [g] =iDelta [f,g] ,.

Note that Δ is a distribution over two arguments, and so, can be smeared as well.

Equivalently, we could have insisted that

:mathcal{T}{ [((partial^mu partial_mu+m^2)phi) [f] ,phi [g] }=-iint d^dx f(x)g(x)

where mathcal{T} is the time ordering operator and that if the supports of f and g are spacelike separated,

: [phi [f] ,phi [g] =0.

See also normal order, Wick's theorem


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • free field — noun : a sound wave field devoid of obstacles causing reflection, refraction, or diffraction …   Useful english dictionary

  • free field overpressure — See free air overpressure …   Military dictionary

  • curl-free field — besūkuris laukas statusas T sritis fizika atitikmenys: angl. curl free field; irrotational field; non circuital field vok. wirbelfreies Feld, n rus. безвихревое поле, n; невихревое поле, n; поле без вихрей, n pranc. champ irrotationnel, m; champ… …   Fizikos terminų žodynas

  • source-free field — solenoidinis laukas statusas T sritis Standartizacija ir metrologija apibrėžtis Vektorinio dydžio, išreikšto kito vektoriaus rotoriumi, laukas. Tokio dydžio divergencija lygi nuliui, o lauko linijos uždaros arba prasideda ir baigiasi jo kraštuose …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • source-free field — solenoidinis laukas statusas T sritis fizika atitikmenys: angl. solenoidal field; source free field vok. quellenfreies Feld, n; solenoidales Feld, n rus. соленоидальное поле, n pranc. champ solénoïdal, m …   Fizikos terminų žodynas

  • Gaussian free field — The Gaussian free field is a lattice subject studied in statistical mechanics.Let Omega be Re, mu Lebesgue measure and let P ( x , y ) be the transition kernel of a random walk on lattice. The Hamiltonian is then given by: H( varphi ) =… …   Wikipedia

  • Free State Pygmy Mouse — Conservation status Least Concern (IUCN 3.1) Scientific classification …   Wikipedia

  • Field Country Homestay — (Хуалянь,Тайвань) Категория отеля: Адрес: No.91 15, Feng Vil., 970 Хуалянь …   Каталог отелей

  • Field emission — (FE) is the emission of electrons from the surface of a condensed phase into another phase due to the presence of high electric fields. In this phenomenon, electrons with energies below the Fermi level tunnel through the potential barrier at the… …   Wikipedia

  • Field-tube boiler — Field tube, sectioned A Field tube boiler (also known as a bayonet tube[1] ) is a form of water tube boiler where the water tubes are single ended. The tubes are closed at one end, and they contain a concentric inner tube. Flow is thus separated… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”