Math 2.0

Quantum theory Definition

Quantum field theory

Builds on Wightman axiomsHilbert spacePoincaré groupUnitary representationTempered distribution

Definition

In the axiomatic (Wightman) sense, a quantum field theory of one real scalar field on four-dimensional Minkowski space is a quadruple (ℋ,U,Ω,ϕ)(\mathcal{H}, U, \Omega, \phi) where:

  1. ℋ\mathcal{H} is a separable complex Hilbert space;
  2. UU is a strongly continuous unitary representation of the identity component of the Poincaré group on ℋ\mathcal{H};
  3. Ω∈ℋ\Omega \in \mathcal{H} is a unit vector, the vacuum;
  4. ϕ\phi assigns to each Schwartz function ff on ℝ4\R^4 an operator ϕ(f)\phi(f), the smeared field. These operators have a common dense domain, and f↦ϕ(f)f \mapsto \phi(f) is an operator-valued tempered distribution;

and these data satisfy the Wightman axioms.

Remarks

Theories with several fields, with fields of nonzero spin, or in other dimensions d≥2d \ge 2 are defined in the same way. Then ϕ\phi is replaced by a finite family of fields that transform under a finite-dimensional representation of the Lorentz group. Other axiom systems describe the same notion from other angles. The Haag–Kastler axioms use nets of operator algebras attached to regions of spacetime. Euclidean axioms use correlation functions on ℝd\R^d.

Example: the free scalar field of mass m>0m > 0 on the bosonic Fock space over L2L^2 of the mass hyperboloid {p:p2=m2, p0>0}\set{p : p^2 = m^2,\ p^0 > 0} satisfies the definition.

Non-example: a field on that Fock space whose operators at spacelike separated points do not commute violates the locality axiom, so it is not a Wightman quantum field theory.

Interacting examples are known in two and three spacetime dimensions, constructed by the methods of constructive quantum field theory; in four dimensions none is known.

Reading path

To understand this page from the ground up, read these 45 pages in order. The foundations come first.

  1. Cross product concept
  2. Determinant concept
  3. General linear group object
  4. Group concept
  5. Hermitian matrix concept
  6. Hilbert space concept
  7. Identity matrix object
  8. Lie group concept
  9. Matrix definition
  10. Möbius transformation concept
  11. Quadratic form concept
  12. Real projective space object
  13. Riemann sphere concept
  14. Semidirect product concept
  15. Signature of a quadratic form concept
  16. Tempered distribution concept
  17. Translation concept
  18. Unitary representation concept
  19. Vector space definition
  20. Wightman axioms concept
  21. Ring definition
  22. Square matrix definition
  23. Transpose definition
  24. Linear map definition
  25. Unit of a ring definition
  26. Conjugate transpose definition
  27. Multilinear map definition
  28. Orthogonal matrix definition
  29. Division ring definition
  30. Alternating multilinear map definition
  31. Orthogonal group object
  32. Unitary matrix definition
  33. The quaternions example
  34. Determinant definition
  35. Unitary group object
  36. Indefinite orthogonal group example
  37. Special linear group object
  38. Special orthogonal group object
  39. Special unitary group SU(n) example
  40. Rotation group SO(3) example
  41. SU(2) example
  42. SL(2,C) example
  43. Lorentz group example
  44. Poincaré group example
  45. Quantum field theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantum_field_theory
About this page
Tag7R9JS3VFW1
URIm2://7R9JS3VFW1
Kinddefinition
WritingW0 stub
MSC81T05, 81T08
DefinesQuantum field theory
UsesWightman axioms, Hilbert space, Poincaré group, Unitary representation, Tempered distribution
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.