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$\newcommand{\real}{\mathrm{real}}$I am having trouble with understanding the axiom (OS3) in this book by Glimm and Jaffe. It defines \begin{equation} \mathcal{A} = \left \{ A(\phi) = \sum_{j = 1}^N c_j \exp \left( \phi(f_j) \right) \; \Big \vert \; c_j \in \mathbb{C}, f_j \in \mathcal{D} \right \} \end{equation} where $\mathcal{D}$ is supposed to be the space of compactly supported, smooth functions on $\mathbb{R}^d$ and $\phi \in \mathcal{D}'(\mathbb{R}^d)$. Now, this is notationally somewhat messed up (imho) but it is also stated earlier in the text, that $\mathcal{D}'(\mathbb{R}^d)$ should refer to real distributions hence, I assume that $\mathcal{D}$ should also refer to real functions - but I have not found this statement anywhere.

They then go on by defining the time-reflection operator $\theta : \mathcal{A} \to \mathcal{A}$ and consider a special supspace $\mathcal{A}_+ \subset \mathcal{A}$ and give the axiom: \begin{equation} 0 \le \int_{\mathcal{D}'(\mathbb{R}^d)} \left( \theta A \right)^- A \mathrm{d} \mu \end{equation} where the superscript $-$ is supposed to mean complex conjugation. This might sound okay but then they state that this is equivalent to the positive definiteness of matrices of the form \begin{equation} M_{i j} = \int_{\mathcal{D}'(\mathbb{R}^d)} \exp \left[ i \phi \left( f_i - \theta f_j \right) \right] \mathrm{d} \mu \left( \phi \right) \end{equation} with $f_j$ in the new space $\mathcal{D}_{\real}(\mathbb{R}^d)$.

I feel that it is completely unclear whether they are talking about $\mathcal{D}$ as a space of real or complex functions. Furthermore, I cannot see the equivalence of the two statements unless $\mathcal{A}$ was defined with an additional $i$ in the exponent and with $\mathcal{D} \supset \mathcal{D}_{\real}(\mathbb{R}^d)$ as a space of real functions.

nlab also gives the almost identical definition as the book but without stating anything about real or complex spaces.

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  • $\begingroup$ That sounds plausible. But how would one obtain the equivalence with the positive definiteness of the matrix? Then $\mathcal{D}_{real}$ would also have to contain complex functions which I really do not believe. $\endgroup$
    – iolo
    Apr 15, 2022 at 6:18
  • $\begingroup$ Sorry, it has been a while since I read their proof. I think what I said before was wrong. I would have to read the proof again in detail to say for sure, but it seems to me that in the definition $\mathcal{A}$ the test functions should be real valued and an $i$ should be included in the exponentials. There is also no reason to speak of $\mathcal{D}_{\rm real}$ as opposed to $\mathcal{D}$. $\endgroup$ Apr 15, 2022 at 13:08
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    $\begingroup$ Their proof is adapted from numdam.org/article/AIHPA_1974__21_4_271_0.pdf which I think is more explicit in having the test functions and distributions real valued. Analyticity helps with finite moment bounds, but they are stated in terms of $\xi f$ with $\xi$ a complex number but $f$ still a real valued test function. $\endgroup$ Apr 15, 2022 at 13:14
  • $\begingroup$ Thank you very much! I will have a look on Monday :) $\endgroup$
    – iolo
    Apr 15, 2022 at 13:55
  • $\begingroup$ From your linked article, it seems clear that the algebra $\mathcal{A}$ should have been defined with an additional $i$ in the exponent. In fact, footnote 2 on p. 93 does note make sense otherwise. Furthermore, we should not have $\mathcal{D}_{\mathrm{real}}(\mathbb{R}^d)$ at all, since everything is real, but rather the set of all test functions that are compactly supported in a set with positive time coordinate. Since your link solved the problem, feel free to write an answer :) $\endgroup$
    – iolo
    Apr 19, 2022 at 9:36

1 Answer 1

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Since this has confused me multiple times, I write this answer in the hope that it might help others.

First, recall that reflection positivity as formulated by Osterwalder and Schrader states that \begin{equation} \sum_{n,m} G_{n+m} \left( \theta f_n^* \otimes f_m \right) \ge 0 \end{equation} for all finite sequences $(f_n)_{n \in \mathbb{N}}$ of complex Schwartz functions on $(\mathbb{R}^d)^n$ with supports in $(\mathbb{R}_{\ge 0} \times \mathbb{R}^{d-1})^n$. Here, \begin{equation} \left( \theta f \right) \left( x^1_1, \dots, x^1_d, x^2_1, \dots x^d_d \right) = f \left( -x^1_1, x^1_2 \dots, x^1_d, -x^2_1, x^2_2 \dots x^2_d, \dots \right) \end{equation} for all $x^1, \dots x^d \in \mathbb{R}^d$.

Second, the whole idea of having a measure $\mu$ on a space of distributions is to be able to write the $G_n$s as moments of that measure. From here on $\mu$ will be a corresponding measure on the space of real distributions. Also, let $Z$ denote the moment-generating function of $\mu$ (assuming it exists) and $\hat{\mu}$ its characteristic function.

Now, for any finite sequences $(\phi_n, c_n)_{n \in \mathbb{N}}$ of real Schwartz functions on $\mathbb{R}^d$ with supports on $\mathbb{R}_{\ge 0} \times \mathbb{R}^{d-1}$ and complex numbers, define \begin{equation} f_a = \frac{1}{a!} \sum_n c_n \phi_n^{\otimes a} \end{equation} for all $a \in \mathbb{N}$. Then a simple calculation (assuming everything is well-defined) shows that \begin{equation} \sum_{m, n} c_n^* c_m Z \left( \theta \phi_n + \phi_m \right) = \sum_{a, b = 0}^\infty G_{a+b} \left( \theta f_a^* \otimes f_b \right) = \lim_{N \to \infty} \sum_{a, b = 0}^N G_{a+b} \left( \theta f_a^* \otimes f_b \right) \ge 0 \, . \end{equation}

Similarly, we obtain a connection with the characteristic function by setting \begin{equation} g_a = \frac{\left(-i\right)^a}{a!} \sum_n c_n \phi_n^{\otimes a} \end{equation} for all $a \in \mathbb{N}$. Then \begin{equation} \sum_{m, n} c_n^* c_m \hat{\mu} \left( \theta \phi_n - \phi_m \right) = \sum_{a, b = 0}^\infty G_{a+b} \left( \theta g_a^* \otimes g_b \right) = \lim_{N \to \infty} \sum_{a, b = 0}^N G_{a+b} \left( \theta g_a^* \otimes g_b \right) \ge 0 \, . \end{equation}

Observe that we need either $Z( \theta \phi_n + \phi_m )$ or $\hat{\mu}( \theta \phi_n - \phi_m )$.

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