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SRQ seminar – September 10th, 2018

Notes from a talk in the SRQ series.

Brydges' lecture on Monday September 10th, 2018. <db5d@math.ubc.ca>

Axioms of Euclidean quantum field theory (EQFT)

I will follow the book [GJ87]. An EQFT is a probability Borel measure $\mu$ on $\mathcal{D}' (\mathbb{R}^d)$ whose Fourier transform

$\displaystyle S (f) = \int_{\mathcal{D}'} e^{i \phi (f)} d \mu, \qquad \text{for } f \in C_0^{\infty} (\mathbb{R}^d),$

satisfies

Analyticity. $S \left( \sum_{i = 1}^n z_k f_k \right)$ is entire in $(z_1, \ldots, z_n) \in \mathbb{C}^n$ for all $f_i \in C_0^{\infty} (\mathbb{R}^d)$, $i = 1, \ldots, n$. This implies existence of the Laplace transform and exponential moments.
Euclidean invariance. The Euclidean group is the set of all continuous bijections of $\mathbb{R}^d$ to itself which preserve Euclidean distance. It contains rotations, translations and reflections and these generate the group (hard to prove). The measure is invariant under all the group. In particular translations in the time direction: $T_s f (y, x_0) = f (y, x_0 - s)$ where $(x_0, x_1, \ldots, x_{d - 1}) = (x_0 \in \mathbb{R}, y \in \mathbb{R}^{d - 1}) \in \mathbb{R}^d$ and reflection: $\theta f (y, x_0) = f (t, - x_0)$.
O.S. (reflection) positivity.
$\displaystyle \mathcal{A} := \{ \text{(complex) span of $e^{\phi (f_1)}, \ldots, e^{\phi (f_n)}$ for all $n$ and all } f_i \in C_0^{\infty} \} .$
Then $\mu$ is OS/reflection positive if
$\displaystyle \int (\theta \bar{F}) F \mathrm{d} \mu \geqslant 0,$
for all $F \in \mathcal{A}_+$ where $\mathcal{A}_+$ is the above span constructed only with $f_i \in C_0^{\infty}$ with support in $\mathbb{R}_+^d = \{(x_0, \ldots, x_{d - 1}) \in \mathbb{R}^d : x_0 > 0\}$. This is the axiom which decides the direction of time.
Ergodicity. If $F \in L^1 (\mu)$ then, pointwise almost surely ($\mu$)
$\displaystyle \frac{1}{t} \int_0^t T_s F \rightarrow \int F \mathrm{d} \mu .$
(This corresponds to uniqueness of the vacuum in relativistic QFT.)
Regularity. with $p \in [1, 2]$,
$\displaystyle |S (f) | \leqslant e^{c (\|f\|_{L^1} +\|f\|_{L^p})} .$

These are the axioms for the simplest kind of EQFT which is called scalar bosonic EQFT. There are more complicated vector-valued bosonic theories with similar axioms. Also there are fermionic theories, which are not measures. These will be discussed later. The terminology O.S. positivity honours [OS73] and [OS75] where this concept was introduced.

Remark 1. For any $F \in L^2 (\mu)$, $\| \theta F\|_{L^2} = \|F\|_{L^2}$ because $\mu$ is Euclidean invariant.

Euclidean invariance and reflection positivity turn out to be difficult to achieve simultaneously. We are about to see that OS positivity enables us to construct a Hilbert space and a semigroup. The Hilbert space connects us to physics by being the Hilbert space of states of a quantum mechanics.

1Reconstruction of quantum mechanics

For $A, B \in \overline{\mathcal{A}_+}$ (closure in $L^2 (\mu)$), two inner products can be defined:

$\displaystyle (A, B)_{L^2 (\mu)} := \int \overline{A (\phi)} B (\phi) \mu (\mathrm{d} \phi) .$ $\displaystyle (A, B)_{\mathcal{H}} := \int \overline{\theta A (\phi)} B (\phi) \mu (\mathrm{d} \phi) .$

Actually, we will later see that the second definition does not satisfy the requirement that $(A, A)_{\mathcal{H}} > 0$ for $A \neq 0$ so we define $\mathcal{N}= \{A \in \overline{\mathcal{A}_+} : (A, A)_{\mathcal{H}} = 0\}$ and then $\mathcal{H} := \overline{\mathcal{A}_+} \setminus \mathcal{N}$ is a Hilbert space. Define a semigroup $P_t : \mathcal{H} \rightarrow \mathcal{H}$ by: for all $t \geqslant 0$ and $A, B \in \mathcal{H}$,

$\displaystyle (A, P_t B)_{\mathcal{H}} = (\theta (A), T_t B)_{L^2 (\mu)} .$

This makes sense since $T_t B \in \mathcal{H}$ if $B \in \mathcal{H}$ and $t \geqslant 0$. On the left hand side $A, B$ represent equivalence classes, but we omit this from our notation and omit the check that our definitions are consistent equality modulo $\mathcal{N}$. This is done carefully in [GJ87].

Claims:

$P_t$ is self–adjoint. To prove this we use unitarity of time translation for the $L^2 (\mu)$ scalar product to get
$\displaystyle (A, P_t B)_{\mathcal{H}} = (\theta A, T_t B)_{L^2 (\mu)} = (T_{- t} \theta A, B)_{L^2 (\mu)} = (\theta T_t A, B)_{L^2 (\mu)} = (P_t A, B)_{\mathcal{H}} .$
$P_t$ is contractive. By self–adjointness
$\displaystyle \|P_t A\|_{\mathcal{H}} = (P_t A, P_t A)_{\mathcal{H}}^{1 / 2} = (A, P_{2 t} A)_{\mathcal{H}}^{1 / 2} \leqslant \|A\|_{\mathcal{H}}^{1 / 2} \|P_{2 t} A\|_{\mathcal{H}}^{1 / 2}$
and by applying the same bound to the right hand side with $t$ replaced by $2 t$ etc. we get
$\displaystyle \|P_t A\|_{\mathcal{H}} \leqslant \|A\|_{\mathcal{H}}^{1 / 2 + 1 / 4} \|P_{4 t} A\|_{\mathcal{H}}^{1 / 4} \leqslant \cdots \leqslant \|A\|_{\mathcal{H}}^{1 / 2 + \cdots + 1 / 2^n} \|P_{2^n t} A\|_{\mathcal{H}}^{1 / 2^n}$
but $\|P_{2^n t} A\|_{\mathcal{H}}^{1 / 2^n} \leqslant \|T_{2^n t} A\|_{L^2}^{1 / 2^n} = \|A\|_{L^2}^{1 / 2^n} \rightarrow 1$ as $n \rightarrow \infty$. Therefore taking the $n \rightarrow \infty$ we conclude that $\|P_t A\|_{\mathcal{H}} \leqslant \|A\|_{\mathcal{H}}$.
$(P_t)_{t \geqslant 0}$ is strongly continuous. Namely $\|P_t A - A\|_{\mathcal{H}} \rightarrow 0$ as $t \rightarrow 0$. (The same technology applies, see [GJ87])

Therefore the Hille–Yoshida theorem ensures the existence of an operator $H$ on $\mathcal{H}$ which is self-adjoint (since $P_t$ is self–adjoint), (unbounded), $\sigma (H) \in [0, \infty)$ and $P_t = e^{- tH}$. By the spectral theorem for self-adjoint operators we can also define the unitary group $U_t = e^{- itH}$. $U_t$ is the time evolution on $\mathcal{H}$ of QM. In [GJ87] this is just part of the construction of a relativistic invariant quantum field theory that satisfies the Wightman axioms, but we will not pursue this. This was the original achievement of [OS73] and [OS75].

2An example

Take a lattice with $5$ points $\{1, \ldots, 5\} \subseteq \mathbb{Z}$. Define $\theta$ as reflection on the middle point. We consider random variables $\phi_1, \ldots, \phi_5$ and the measure

$\displaystyle \mathrm{d} \mu (\phi) = \frac{1}{Z} e^{- \frac{1}{2} \sum_{i = 0}^5 (\phi_{i + 1} - \phi_i)^2} \prod_{i = 1}^5 \rho (\mathrm{d} \phi_i)$

where we assume $\phi_0 = \phi_6 = 0$ and where $\rho$ is a finite measure. An example is $\rho = \delta_{+ 1} + \delta_{- 1}$, which gives the Ising model in $d = 1$ with five points.

What is the meaning of $\mathcal{N}$? We have $\mathcal{A}_+ = \{A (\phi) = A (\phi_3, \phi_4, \phi_5)\}$. Let $(\iota A) (\phi_3) =\mathbb{E}_{\mu} (A| \phi_3)$. This maps $\overline{\mathcal{A}_+}$ into the subspace of $L^2 (\mu)$ of r.v. measurable wrt $\phi_3$. For $A, B \in \mathcal{A}_+$

$\displaystyle (A, B)_{\mathcal{H}} =\mathbb{E}_{\mu} (\theta (\bar{A}) B) =\mathbb{E}_{\mu} [\mathbb{E}_{\mu} ((\theta \bar{A}) B| \phi_3)]$ $\displaystyle =\mathbb{E}_{\mu} [\mathbb{E}_{\mu} (\theta \bar{A} | \phi_3)\mathbb{E}_{\mu} (B| \phi_3)] =\mathbb{E}_{\mu} \left[ \overline{(\iota A) (\phi_3)} \hspace{0.17em} \iota B (\phi_3) \right]$

This shows that $\mu$ is reflection positive and that $\mathcal{N}= \left\{ A \in \mathcal{A}_+ | \iota A = 0 \hspace{0.17em} \mu \text{--a.s.} \right\}$ is the kernel of $\iota$ so

$\displaystyle \overline{\mathcal{A}_+} \setminus \mathcal{N} \simeq \{F (\phi_3) \in L^2 (\mu)\} .$

So $\mathcal{N}$ is a quite big space which reduces $\overline{\mathcal{A}_+}$ to a space of functions of one variable.

OS positive measures are very special. They were studied in detail in [FILS78]. For example the fractional Laplacian (energy of the form $(\phi, (- \Delta)^{1 / 2} \phi)$) is OS positive, but not energies of the form $(\Delta \phi, \Delta \phi)$ (elasticity equation). For lattice models one can have OS positivity for reflections through hyperplanes that pass through lattice points or for reflections through hyperplanes that pass between lattice points. I think that $(\phi, (- \Delta)^{1 / 2} \phi)$) only has OS positivity for the latter.

Problem: For the Ising model, consider the weak limit of $\mu$ as the number of points go to infinity (start by assuming it exists).

Find $\mathcal{H}$, $P_t : \mathcal{H} \rightarrow \mathcal{H}$ with $t = 0, 1, 2, \ldots$
Determine how $\int \phi_0 \phi_n \mathrm{d} \mu$ decays as $n \rightarrow \infty$.

This is a $\mathbb{Z}$-QFT (invariant under the automorphisms of $\mathbb{Z}$).

Bibliography

[OS75]: Konrad Osterwalder and Robert Schrader. Axioms for Euclidean Green's functions. II. Commun. Math. Phys., 42:281–305, 1975. With an appendix by Stephen Summers.
[OS73]: Konrad Osterwalder and Robert Schrader. Axioms for Euclidean Green's functions. Commun. Math. Phys., 31:83–112, 1973.
[GJ87]: J. Glimm and A. Jaffe. Quantum Physics, A Functional Integral Point of View. Springer, Berlin, 2nd edition, 1987.
[FILS78]: J. Fröhlich, R. Israel, E.H. Lieb, and B. Simon. Phase transitions and reflection positivity. I. General theory and long range lattice models. Commun. Math. Phys., 62:1–34, 1978.