Examples of cts limit systems

What is a quantum field state.tex

@@ -200,15 +200,15 @@ \section{What is a field theory}\label{sec:whatisqft}
 
 \section{Effective field theories}\label{sec:effectivetheories}
 
- Suppose we have some extraordinarily complicated system of many particles: a good example to keep in mind is \emph{water}. Now if it were easy for us to make any conceivable measurement allowed by quantum mechanics of the system \emph{at no cost}, then there is \emph{no way} we'd be fooled into thinking water is anything other than a collection of a vast number of fundamental particles, quarks, gluons, etc., in some incredibly complicated evolving entangled state. The reason we don't see water like this is that we \emph{can't} make any measurement of the system without paying some kind of bill: the more complicated the measurement the more we have to pay. Thus we have to settle with making measurements of simpler quantities. For example, our eyes are basically a pair of pretty crappy photon detectors and thus when we look at a water sample we are simply carrying out a very noisy and inefficient POVM. Now here is the main point: when you only have access to a handful of observables then you can formulate a \emph{simpler hypothesis} which can still explain all the observational data you can obtain. This simpler hypothesis is an \emph{effective theory} for the system. Simpler here can mean many things, but in the context of this paper it is via a field theory\footnote{Why are fields simple? The answer is calculus: it is often easier to calculate integrals than sums.}.
+Suppose we have some extraordinarily complicated system of many particles -- a good example to keep in mind is \emph{water}. Now if it were easy, \emph{at no cost}, for us to make any conceivable measurement on the system allowed by quantum mechanics, then there is \emph{no way} we'd be fooled into thinking water is anything other than a collection of a vast number of fundamental particles, quarks, gluons, etc., in some incredibly complicated evolving entangled state. The reason we don't see water like this is that we \emph{can't} make any measurement of the system without paying some kind of bill: the more complicated the measurement, the more we have to pay. Thus we have to settle with making measurements of simpler quantities. For example, our eyes are basically a pair of pretty crappy photon detectors and thus when we look at a water sample we are simply carrying out a very noisy and inefficient POVM. Now here is the main point: when you only have access to a handful of observables then you can formulate a \emph{simpler hypothesis} which can still explain all the observational data you can obtain. This simpler hypothesis is an \emph{effective theory} for the system. Simpler here can mean many things, but in the context of this paper it is via a field theory\footnote{Why are fields simple? The answer is calculus: it is often easier to calculate integrals than sums.}.
 
-A convenient way to model the large-scale degrees of freedom that we humans with our limited resources can access is by \emph{zooming out}. How can one, in general, implement the operation of ``zooming out''? Since zooming out corresponds to ignoring information the answer is via an irreversible CP map $\mathcal{E}$. The reason that it has to be irreversible is that it must prevent us from measuring degrees of freedom that we would otherwise be able to measure: after all, if we could measure all the observables after zooming out that we could measure before then in what sense can we be have said to have zoomed out? In the context of lattice systems there is a very convenient way to implement the zoom-out operation, namely, via Kadanoff blocking. This is the CP map whereby a block of spins is mapped to single spin via the partial trace channel, and then the lattice is rescaled.
+How can we model the large-scale degrees of freedom that we humans with our limited resources can access? One very simplified way is by developing a \emph{zooming out} operation. Since zooming out corresponds to \emph{ignoring information}, this operation should be representable in quantum mechanics as an irreversible CP map $\mathcal{E}$. The reason that it has to be irreversible is that it must prevent us from measuring degrees of freedom that we would otherwise be able to measure: after all, if we could measure all the observables after zooming out that we could measure before then in what sense can we be have said to have zoomed out? In the context of lattice systems there is a very convenient way to implement the zoom-out operation, namely, via Kadanoff blocking. This is the CP map whereby a block of spins is mapped to single spin via the partial trace channel, and then the lattice is rescaled.
 
-Suppose we have some tremendously complicated microscopic system pertaining to degrees of freedom at some fundamental length scale $\Lambda$ in some state $\rho_\Lambda$. However, we can only perform limited measurements at our terrestrial length scale $\sigma$ modelled by effects of the form $\mathcal{E}_{\sigma}(E)$, where $E \in \mathcal{A}_{\sigma}$ is an effect in the space of \emph{our} observables. Since we can only measure a handful of all the possible observables $\mathcal{A}_\Lambda$ on the microscopic theory, we are satisfied with the explanation provided by any \emph{effective state} $\rho_{\text{eff}}$ which looks indistinguishable from $\rho_\Lambda$ according to any measurement $\mathcal{E}_{\sigma}(E)$, i.e.\ any state obeying
+Suppose we have some tremendously complicated microscopic system of degrees of freedom at some fundamental length scale $\Lambda$ in some state $\rho_\Lambda$. However, we can only perform limited measurements at our terrestrial length scale $\sigma$ modelled by effects of the form $\mathcal{E}_{\sigma}(E)$, where $E \in \mathcal{A}_{\sigma}$ is an effect in the space of \emph{our} observables. Since we can only measure a handful of all the possible observables $\mathcal{A}_\Lambda$ on the microscopic theory, we are satisfied with the explanation provided by any \emph{effective state} $\rho_{\text{eff}}$ which looks indistinguishable from $\rho_\Lambda$ according to any measurement $\mathcal{E}_{\sigma}(E)$, i.e.\ any state obeying
 \begin{equation}
 	\tr(\mathcal{E}_{\sigma}(E) (\rho_\Lambda - \rho_{\text{eff}})) \approx 0, \quad \text{for all } E\in \mathcal{A}_\sigma.
 \end{equation}
-The fewer the observables we can measure, the simpler we can take $\rho_{\text{eff}}$ to be. In the case of fields, we are only able to measure a handful of observables $E(x)$ indexed by some continuous label $x$; we'll explain this a bit more concretely in the following sections.
+The fewer the observables we can measure, the simpler our hypothesis for $\rho_{\text{eff}}$ can be. In the case of fields, we are only able to measure a handful of observables $E(x)$ indexed by some continuous label $x$; we'll explain this a bit more concretely in the following sections.
 
 One aspect of this discussion may be puzzling for readers familiar with quantum field theories: why are we insisting on saying a change of scale is a lossy operation when, e.g., in CFTs a scale change is a \emph{reversible unitary operation}? The answer is that, in terms of a \emph{microscopic theory} with a cutoff, zooming out \emph{must} be a lossy operation as we can push degrees of freedom past the cutoff. However, in terms of an effective theory, a scale change can be unitary because there is an effective decoupling of the large-scale  and the small-scale degrees of freedom and the action of a finite scale change doesn't, \emph{in the large cutoff limit}, couple the different sets of degrees of freedom. 
 
@@ -301,7 +301,7 @@ \subsection{Large-scale observables}
 
 \section{Two examples}
 
-To illustrate the components of the Wilsonian formulation we'll specifically refer to two important examples 
+To illustrate the components of the Wilsonian formulation we'll specifically refer to two important examples. The first example is a standard choice in QFT, namely that of the (hard or smooth) momentum cutoff. We briefly sketch the data required for the Wilsonian formulation in this case and connect it with the traditional description of perturbative interacting QFT by specifying a ``zooming-out'' CP map. The second example we consider, namely, the real-space cutoff requires a little more care to put it into a form appropriate for the Wilsonian formulation: here we encounter additional problems because we need to compare lattices with different lattice spacing. We show how Kadanoff blocking supplies us with a solution, and further how to use introduce a ``zooming-out'' CP map describing the large-scale observables. 
 
 
 \subsection{The momentum cutoff}
@@ -310,7 +310,7 @@ \subsection{The momentum cutoff}
 Key points to stress: we automatically have a way to compare two states with different cutoff.
 
 \subsection{The real-space cutoff}
-The real-space cutoff is implemented by putting the system on a lattice. In contrast to the momentum cutoff case we have a new problem: how to compare the states of two different lattices. 
+The real-space cutoff is implemented by putting the system on a lattice with lattice spacing $a$. In discussing the real-space cutoff case we encounter an additional problem which we did not (directly) have to engage with in the momentum cutoff case: how do we compare two lattices with different lattice spacing? 
 
 \section{Quantum field states via completion}\label{sec:qftcompletion}
 Let's tell a story about how physicists on a world far far away might have invented the real numbers $\mathbb{R}$.  The scientists of this world wanted to measure distances using rulers with tick marks spaced at regular intervals. On this world it was agreed that all rulers had to have their ticks spaced by fractions of some standard tick length $a_0$ (the length of the emperor's foot). Thus the tick spacing $\epsilon a_0$ of a ruler was required to be a positive rational ratio $\epsilon \in \mathbb{Q}^+$ of the standard tick length $a_0$. Owing to its canonical nature $a_0$ was set to $1$ by convention. To the scientists of this world all lengths were rational numbers.