A small puzzle on CFTs

Let us consider a unitary Lorentzian CFT with a hermitian operator

O (x)

. We know that

at the origin for the dilation operator

D

. If we impose that the scaling dimension

Δ

is real, then the right hand side is real hence we need

D

to be antihermitian, i.e.

D^{†} = - D

. This makes sense as

e^{λD}

is unitary for real scaling

λ

only if

D

is antihermitian.

Let us now consider the action of

D

on Hilbert space. As

D

annihilates vacuum, we have

showing that

ϕ (0) vacuum ⟩

needs to be an eigenfunction of the dilation operator

D

with eigenvalue

Δ

. However, this implies that

D

is a hermitian operator because only hermitian operators have real eigenvalues.¹

Resolution: There is actually no paradox because

D

does not act on

ϕ (x) vacuum ⟩

, because it does not define a state in Hilbert space!

In Lorentzian signature, the operator valued fields

ϕ (x)

are not well-defined functions, but rather so-called tempered distributions: they define vectors in the Hilbert space only after they are being smeared with a Schwartz function:

¹This is easy to show: consider an inner product of two operators $A$ and $B$ where the product $(A, B)$ is antilinear in its first argument and linear in the second. We define the hermitian conjugate of an operator $D$ by the equality $(D^{†} A, B) = (A, DB)$ . Thus $DA = λA$ implies $D^{†} A = λ^{*} A$ :

(D^{†} A, A) = (A, DA) = (A, λA) = λ (A, A) = (λ^{*} A, A)

(4.3)

hence $D^{†} = \pm D \to λ^{*} = \pm λ$ for $λ^{*}$ being the complex conjugate of $λ$ . This concludes the proof.

4.2 A small puzzle on CFTs