Quantum theory Axiomatic and algebraic quantum field theory Theorem

CPT theorem

Builds on Wightman axioms Lorentz group Poincaré group

Statement

Let a quantum field theory on Minkowski space $\R^{1,3}$ satisfy the Wightman axioms, with vacuum $\Omega$ . Let $U(a, \Lambda)$ be its representation of the covering group of the proper orthochronous Poincaré group. Assume the normal commutation relations at spacelike separation. That is, two fields anticommute if both have half-integer spin, and commute otherwise. Then there is an antiunitary operator $\Theta$ on the Hilbert space such that:

$\Theta \Omega = \Omega$ ;
$\Theta U(a, \Lambda) \Theta^{-1} = U(-a, \Lambda)$ for all $(a, \Lambda)$ ;
$\Theta$ maps each field $\phi(x)$ to its adjoint at $-x$ , up to a constant phase depending on the Lorentz type of the field. For a Hermitian scalar field, $\Theta \phi(x) \Theta^{-1} = \phi(-x)$ .

Remarks

The idea of Jost's proof: the Wightman functions extend analytically to the forward tube and are covariant under the complex Lorentz group. The proper complex Lorentz group contains $-1$ , which is not in the real proper orthochronous group. At real points of the extended tube (Jost points), where all differences are spacelike, locality lets one reverse the order of the fields. Combining the two gives an identity between Wightman functions at $x$ and at $-x$ in reversed order. This identity is exactly the statement that $\Theta$ defined on states is antiunitary and has the stated properties.

Consequences: a particle and its antiparticle have the same mass and the same lifetime. Jost also showed that CPT invariance is equivalent to weak local commutativity, a condition on Wightman functions at Jost points.

Going deeper

For a student

In the Wightman framework the theorem needs only Lorentz invariance, positivity of the energy and locality. It sits next to the spin–statistics theorem, which uses the same analytic continuation. A measured difference between the masses of a particle and its antiparticle would refute local relativistic quantum field theory, and such experiments are standard tests.

Reading path

To understand this page from the ground up, read these 42 pages in order. The foundations come first.

Cross product concept
Determinant concept
General linear group object
Group concept
Hermitian matrix concept
Identity matrix object
Lie group concept
Matrix definition
Möbius transformation concept
Quadratic form concept
Real projective space object
Riemann sphere concept
Semidirect product concept
Signature of a quadratic form concept
Translation concept
Vector space definition
Wightman axioms concept
Ring definition
Square matrix definition
Transpose definition
Linear map definition
Unit of a ring definition
Conjugate transpose definition
Multilinear map definition
Orthogonal matrix definition
Division ring definition
Alternating multilinear map definition
Orthogonal group object
Unitary matrix definition
The quaternions example
Determinant definition
Unitary group object
Indefinite orthogonal group example
Special linear group object
Special orthogonal group object
Special unitary group SU(n) example
Rotation group SO(3) example
SU(2) example
SL(2,C) example
Lorentz group example
Poincaré group example
CPT theorem statement

Tag	`EV1TZJ0KTM`
URI	`m2://EV1TZJ0KTM`
Kind	statement, theorem
Writing	W0 stub
MSC	81T05, 81R05
Status	This copy of the site has no build data. Run `m2 status EV1TZJ0KTM`.
About	Quantum field theory
Uses	Wightman axioms, Lorentz group, Poincaré group
Changed	3 November 2026