# On the $\ast$ map in the Osterwalder-Schrader axioms

I'm studying "A Mathematical Introduction to Conformal Field Theory" by Schottenloher and there is one point on the Osterwalder-Schrader axioms that I am a bit confused about. They are phrased in the book as follows:

We start with a countable index set $$B_0$$ and consider multi-indices $$(i_1,\dots, i_n)\in B_0^n$$. We further denote $$B = \bigcup_{n=0}^\infty B_0^n$$. We have functions $$G_{i_1\dots i_n}:M_n\to \mathbb{C}$$, where $$M_n$$ is the space of all $$(z_1,\dots,z_n)\in \mathbb{C}^n$$ with $$z_i\neq z_j$$. These functions are to be interpreted as Green's functions and demanded to obey

1. Locality: The Green's functions are permutation invariant, i.e., for any $$\sigma\in S_n$$ the permutation group in $$n$$ letters we have $$G_{i_{\sigma(1)}\dots i_{\sigma(n)}}(z_{\sigma(1)},\dots, z_{\sigma(n)})=G_{i_{1}\dots i_{n}}(z_{1},\dots, z_{n}).$$

2. Covariance: For any isometry of the plane $$w(z)$$ there are numbers $$(h_i,\bar h_i)$$ such that $$G_{i_1\dots i_n}(z_1,\bar z_1,\dots, z_n,\bar z_n)=\prod_{j=1}^n\left(\dfrac{dw}{dz}(z_j)\right)^{h_j}\left(\overline{\dfrac{d w}{dz}(z_j)}\right)^{\bar h_j}G_{i_1\dots i_n}(w_1,\bar w_1,\dots, w_n,\bar w_n).$$

3. Reflection Positivity: There is a map $$\ast : B_0\to B_0$$ with $$\ast^2=1$$ and a continuation $$\ast:B\to B$$ such that $$G_{i_1,\dots,i_n}(z)=G_{(i_1,\dots,i_n)^\ast}(-z^\ast),$$ and such that $$\langle \underline{f},\underline{f}\rangle \geq 0$$ for all $$\underline{f}\in \underline{\mathscr{S}}^+$$ where $$\langle \underline{f},\underline{f}\rangle = \sum_{i,j\in B}\sum_{n,m}\int_{M_{n+m}}G_{i^\ast j}(-z_1^\ast,\dots,-z_n^\ast,w_1,\dots, w_m)f_i(z)^\ast f_j(w) d^n z d^m w.$$ Here $$\underline{\mathscr{S}}^+$$ is the space of all sequences $$(f_i)_{i\in B}$$ where $$f_i\in \mathscr{S}_n^+$$ when $$i\in B_0^n$$ with only a finite number of $$f_i\neq 0$$. $$\mathscr{S}_n^+$$ is the space of Schwarz functions on $$M_n$$ supported in $$M_n^+$$, where the later is defined as the subspace of $$M_n$$ in which $${\rm Re}(z_i)>0$$ for all $$i=1,\dots, n$$.

Now, what I'm confused about is that $$\ast$$ map. It seems it should play the role of the adjoint, but I'm confused on why it is defined in the way it is defined. In fact, I still find it confusing for $$\ast$$ to act on the indices, but I feel this is because one is thinking of $$(i_1,\dots, i_n)$$ as associated to a particular field $$\Phi_{i_1\dots i_n}$$. In any case:

1. What is the motivation to define this $$\ast$$ map in the way it is defined, i.e., as a map $$\ast : B_0\to B_0$$ with $$\ast^2=1$$ such that there is a continuation $$\ast:B\to B$$ for which $$G_{i_1,\dots,i_n}(z)=G_{(i_1,\dots,i_n)^\ast}(-z^\ast)$$?

2. What is the relation of $$\ast$$ with the adjoint in the Hilbert space?

3. Is this $$\ast$$ related to the $$\ast$$ map that appears in the $$\ast$$-algebra approach to axiomatic quantum field theory? It seems they are related, but the formalisms seem a bit distinct.

• Could the $*$ acting in the index set mean that you replace the fields in the Green functions by their adjoints? That makes sense as the usual definition of reflection positivity Commented Dec 27, 2023 at 17:39
• I do think that $\Phi_{i^\ast}$ should be interpreted as the adjoint of $\Phi_i$, but I'm struggling to see that the given definition does the job. For example, in the $\ast$-algebra approach the $\ast$ operation is further demanded to obey $$(x+y)^\ast = x^\ast +y^\ast,\quad (\lambda x)^\ast = \bar \lambda x^\ast,\quad (xy)^\ast = y^\ast x^\ast,$$ for all $x,y\in \mathfrak{A}$ and $\lambda\in\mathbb{C}$. I don't see analogues of these here. I'm also a bit confused why $G_{i_1,\dots,i_n}(z)=G_{(i_1,\dots,i_n)^\ast}(-z^\ast)$ characterizes $\ast$ as the adjoint.
– Gold
Commented Dec 27, 2023 at 18:15
• Reflection positivity involves the inner products of states $f$ defined, in this case, on the $y$ axis. (Note that the $z\to -z^*$'s is reflection in the $y$ axis). Your last equation $(xy)^*= y^*x^*$ applies to operators not states. Commented Dec 27, 2023 at 18:20
• I understand that the equations I wrote apply to operators. Maybe I could rephrase the question as: given that under the Hilbert space reconstruction $G_{i_1\dots i_n}(z_1,\dots, z_n) = \langle \Omega|\Phi_{i_1}(z_1)\cdots \Phi_{i_n}(z_n)|\Omega\rangle$, how does this $\ast$ relates to the Hilbert space adjoint $\dagger$? I'm trying to compute the adjoint defined by this inner product $\langle \underline{f},\underline{g}\rangle$ to see if I understand it...
– Gold
Commented Dec 27, 2023 at 19:46

The $$*$$ is not the adjoint. It is the operation of reflection in the $$y$$ axis.

I would change the wording so that $$"G_{i_1,\dots,i_n}(z)=G_{(i_1,\dots,i_n)^\ast}(-z^\ast)",$$ becomes $$*:G_{i_1,\dots,i_n}(z)\mapsto G_{(i_1,\dots,i_n)^\ast}(-z^\ast).$$

O and S define the inner product of a state $$|f\rangle \stackrel{\rm def}{=} \int_{x^0_i>0} d^dx_1 \cdots d^d x_n \Phi(x_1\ldots \Phi(x_n)|0\rangle f(x_1)f(x_2)\cdots f(x_n)$$ (where the $$f$$'s are supported on the positive time region) with itself as the correlator $$\int_{x^0_i>0; \tilde x_i^0<0} dx_1 \ldots f(\tilde x_1) \cdots f(\tilde x_n) \langle\Phi^*_n(\tilde x_n) \cdots\Phi^*(\tilde x_1)\Phi(x_1) \Phi(x_n)\rangle f(x_1) \cdots f(x_n).$$ where each $$\tilde x$$ is located at the mirror image, due to reflection about the $$x^0=0$$ plane, of the corresponding $$x$$ point. Reflection positivity is the statement that this correlator is positive. This construction defines the vacuum state $$|0\rangle$$ as the object obtained this way when $$n=0$$. In other words the numerical values of the correlators define the Hilbert space in much the same way as the GNS construction in an abstract C$$^*$$ algebra defines a Hilbert space on which the algebra acts.

Under this inner product the complex variable $$\Phi(x)$$ that appears in a path integral becomes an operator whose adjoint $$\Phi^\dagger$$ is given by the $$\Phi^*(x)$$.

Oops: I left out the statement that in the definition of the state, the $$x_i$$ should be time ordered. That condition does not matter for the correlator as the correlator automatically corresponds to time-ordered fields in the opertor language.

• Thanks! I think I was led away by the notation and thought that $\ast$ had something to do with the adjoint. This makes more sense now. The only thing is that the book does indeed say that the Green's functions are supposed to obey $G_i(z)=G_{i^\ast}(-z^\ast)$. Isn't this usually required in the OS axioms? Maybe it is a mistake in the book? I have found another today, in the Conformal Ward Identities, so maybe this is another issue...
– Gold
Commented Dec 27, 2023 at 23:14
• I do not understand the $G(z)= G(-z^*)$ statement, but this might be my failure. I think there is a wikipedia article that might be clearer:en.wikipedia.org/wiki/Schwinger_function#Reflection_positivity Commented Dec 27, 2023 at 23:16