CPT Symmetry as a Geometric Theorem from the T-Boundary

The CPT theorem is one of the most fundamental results in quantum field theory. It states that any Lorentz-invariant local quantum field theory is invariant under the combined operation of charge conjugation (C: particles ↔ antiparticles), parity reversal (P: spatial reflection), and time reversal (T: t → −t). First proved by Pauli and Lüders in the 1950s and given its rigorous axiomatic form by Jost, the CPT theorem explains why the matter-antimatter asymmetry of the universe cannot arise from a purely local, Lorentz-invariant quantum field theory — CPT ensures perfect symmetry between particles and their antiparticles in all physical processes.

Yet the standard proof of the CPT theorem is more mysterious than it appears. It relies on the following chain of reasoning: Lorentz invariance requires that the product CPT can always be represented by a rotation of 2π in the complexified Lorentz group SO(3,1;C). Since this rotation is in the connected component of the identity, it is the identity on physical states. Therefore CPT = 1 in every Lorentz-invariant theory. The CPT theorem is, in this sense, not a dynamical statement but an analytic continuation argument. It works because the complexified Lorentz group has a specific topology that makes CPT indistinguishable from a rotation.

The Hidden Geometric Assumption

The analytic continuation from Minkowski to Euclidean space — implicit in every proof of the CPT theorem — requires that the Wightman functions extend to analytic functions on a "permuted extended tube" in complexified Minkowski space. This is not a consequence of Lorentz invariance alone; it is a condition on the boundary behaviour of the theory. Specifically, it requires that the theory be well-defined on the conformal compactification of Minkowski space, including null infinity ℐ^± and spatial infinity i⁰.

The Shadow Framework makes this explicit. The conformal boundary ℐ⁰ separates the forward-time sector M⁴₊ from its T-image M⁴₋. The T-boundary condition identifies the theory in M⁴₊ with the theory in M⁴₋ via the shadow symmetry Δ ↔ 2−Δ. This identification is precisely the CPT operation:

Decomposing CPT into Its Three Components

Time reversal (T) corresponds to the reflection through the conformal boundary: a state in the forward-time sector maps to its partner state in the T-image sector. This is the geometric T-reflection τ → −τ at conformal time τ = 0.

Parity (P) arises from the orientation reversal implicit in the boundary identification. The conformal boundary ℐ⁰ is a three-dimensional null hypersurface, and the induced orientation on ℐ⁰ from M⁴₊ is opposite to that from M⁴₋. This orientation reversal is the parity operation in three spatial dimensions.

Charge conjugation (C) arises from the complex conjugation implicit in the shadow pairing. The shadow partner of an operator O_Δ in the holomorphic sector lives in the antiholomorphic sector; complex conjugation maps the holomorphic to the antiholomorphic sector. For charged fields, this complex conjugation exchanges positive and negative frequency modes, which is exactly charge conjugation.

Why CPT Cannot Be Violated Without Breaking the Boundary

In the standard treatment, CPT violation would require a violation of Lorentz invariance or non-locality. In the Shadow Framework, CPT violation would require that the two-sided manifold M⁴₊ ∪  M⁴₋ be ill-defined — that there be no consistent boundary condition at ℐ⁰ relating the two sectors. This would mean the universe has no well-defined conformal boundary, which is equivalent to the statement that null infinity does not exist as a smooth manifold. Since the existence of ℐ^± as a smooth manifold is equivalent to the peeling property of gravitational radiation (the Bondi-van der Burg-Metzner-Sachs theorem), CPT violation would imply that gravitational waves do not peel — a testable prediction that is strongly constrained by LIGO observations.

CPT is not a symmetry imposed on quantum field theory from outside. It is the boundary condition that makes the universe geometrically consistent: the requirement that the forward-time sector and its T-image fit together smoothly at the conformal boundary.

This perspective explains why CPT appears to hold universally: not because it is a postulate, but because its violation would require the conformal structure of spacetime — the very structure that organizes gravitational radiation, scattering amplitudes, and holography — to be ill-defined. CPT is as fundamental as the existence of null infinity itself.