Quantum field theory
Feynman diagram
History
Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity Gauge theory Yang–Mills theory
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Noether charge Topological charge
Tools Anomaly Background field method BRST quantization Correlation function Crossing Effective action Effective field theory Expectation value Feynman diagram Lattice field theory LSZ reduction formula Partition function Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations
Standard Model Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism
Incomplete theories String theory Supersymmetry Technicolor Theory of everything Quantum gravity
Scientists Adler Anderson Anselm Bargmann Becchi Belavin Bell Berezin Bethe Bjorken Bleuer Bogoliubov Brodsky Brout Buchholz Cachazo Callan Coleman Dashen DeWitt Dirac Doplicher Dyson Englert Faddeev Fadin Fayet Fermi Feynman Fierz Fock Frampton Fritzsch Fröhlich Fredenhagen Furry Glashow Gelfand Gell-Mann Goldstone Gribov Gross Gupta Guralnik Haag Heisenberg Hepp Higgs Hagen 't Hooft Iliopoulos Ivanenko Jackiw Jona-Lasinio Jordan Jost Källén Kendall Kinoshita Klebanov Kontsevich Kuraev Landau Lee Lehmann Leutwyler Lipatov Łopuszański Low Lüders Maiani Majorana Maldacena Migdal Mills Møller Naimark Nambu Neveu Nishijima Oehme Oppenheimer Osterwalder Parisi Pauli Peskin Plefka Polyakov Pomeranchuk Popov Proca Rubakov Ruelle Salam Schrader Schwarz Schwinger Segal Seiberg Semenoff Shifman Shirkov Skyrme Sommerfield Stora Stueckelberg Sudarshan Symanzik Thirring Tomonaga Tyutin Vainshtein Veltman Virasoro Ward Weinberg Weisskopf Wentzel Wess Wetterich Weyl Wick Wightman Wigner Wilczek Wilson Witten Yang Yukawa Zamolodchikov Zamolodchikov Zee Zimmermann Zinn-Justin Zuber Zumino
v t e

In physics a free field is a field without interactions, 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 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.

Canonical Commutation Relation

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 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

${\displaystyle \partial ^{\mu }\partial _{\mu }\phi +m^{2}\phi =0}$.

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

Then,

${\displaystyle \{\phi (x),\phi (y)\}=\Delta (x;y)}$

where here, φ is a classical field and {,} is the Peierls bracket.

Then, the canonical commutation relation is

${\displaystyle [\phi [f],\phi [g]]=i\Delta [f,g]\,}$.

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

Equivalently, we could have insisted that

${\displaystyle {\mathcal {T))\{[((\partial ^{\mu }\partial _{\mu }+m^{2})\phi )[f],\phi [g]]\}=-i\int d^{d}xf(x)g(x)}$

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

${\displaystyle [\phi [f],\phi [g]]=0}$.