\section{Introduction}

Holography is a correspondence between apparently very different physical systems. A fruitful way to think of this correspondence is that it encodes the renormalization group (RG) flow of a (generally strongly coupled) quantum field theory into a geometric system, generally in higher dimension, that under certain circumstances may be classical or semi-classical. Conformal fixed points of the field theory's RG flow possessing a conformal group $G$ correspond to systems on geometries whose isometry coincides with $G$, at least locally. Although first introduced as a duality between strongly coupled large $N$ field theories and classical gravitational theories with matter, it is known that the correspondence runs more deep. In fact, free $O(N)$ vector models truncated to the single trace sector are now known to be dual to semi-classical theories with an enormous gauge symmetry, generally known as higher spin theories \cite{Klebanov:2002ja, Sezgin:2002rt, Leigh:2003gk, Giombi:2009wh, Giombi:2012ms}. In Refs. \cite{Leigh:2014tza,Leigh:2014qca}, it was argued that the gauge symmetry of the higher spin theories has an origin in a bi-local, linear symmetry of free field theories, and employing the Polchinski exact renormalization group (ERG) methods, a dual canonical bulk theory was derived. 
In this description, the bulk fields correspond to a connection 1-form for the bi-local symmetry plus an adjoint-valued 0-form. There is a solution of the bulk theory in which the adjoint 0-form field vanishes and the connection is taken to be flat; this solution corresponds to the conformal fixed point, and the flat connection encodes the geometry associated with that fixed point (e.g., $AdS_{d+1}$ when the fixed point has relativistic $d$-dimensional symmetry).

This holographic description is in fact effective: many of the usual elements of the AdS/CFT dictionary can be seen to emerge from the above construction. In particular, there is an exact action for the bulk higher spin theory\footnote{More precisely, the bulk dynamics is described in the Hamiltonian formalism, which can then be rewritten in terms of a phase space action.} and it has been shown explicitly that the action when evaluated on-shell precisely generates all of the correlation functions of the field theory. The `holographic dictionary' of this higher spin theory contains many familiar entries, along with several features (such as the structure of `Witten diagrams') which are somewhat different, but appropriate to the first order connection formalism. Furthermore, it has been shown \cite{Jin:2015aba} that at the linearized order, the bulk equations of motion in this higher spin theory are canonically equivalent to the Fronsdal equations, which is a basic sanity check from the point of view of group theory. 

The story thus far has focussed on the generating functional of correlators in the vacuum state, which the original GKPW dictionary was built around. The goal of the present paper is to extend the ERG analysis to the renormalization group flows of the \emph{wave functionals} of the vacuum and a class of excited states; see  \cite{Symanzik:1981wd, Luscher:1985iu, Cooper:1987pt, D.Minic:1996aa} for previous work along these lines. For simplicity, we will confine ourselves in this paper to the $O(N)$-singlet sector of the bosonic vector model, with the excited states created by the action of single-trace operators acting on the vacuum. The main result which will emerge from this analysis is a formulation of ERG for states in terms of a continuum analogue of a \emph{tensor network}. For instance, the ERG flow equation for the vacuum state takes the form 
\begin{eqnarray} 
z\frac{\partial}{\partial z}\Big|\Omega(z)\Big\rangle =  i\Big(\boldsymbol{K}(z) + \boldsymbol{L}(z)\Big) \Big|\Omega(z)\Big\rangle,
\end{eqnarray}
where both $\boldsymbol{K}$ and $\boldsymbol{L}$ are local (in position space) unitary operators; a similar equation works out for excited states as well. This unitary action can be naturally extended to $O(N)$-singlet excited states by requiring as an RG principle that these states, as well, have unitary flows.  This requirement is, in fact, equivalent to requiring that the sources preparing the states flow according to their standard beta functions. The action of the operator $\boldsymbol{K}$ on the state has the effect of freezing out the high-energy modes while leaving the low-energy modes untouched. For excited states, this has an interesting interpretation in terms of entanglement -- the operator $\boldsymbol{K}$ removes all entanglement between high-energy and low-energy modes. In other words, $\boldsymbol{K}$ acts as a disentangler in \emph{momentum space}. This is to be contrasted with the usual description of tensor networks such as MERA \cite{Vidal:2007hda, Pfeifer:2008jt} or cMERA \cite{Haegeman:2011uy, Nozaki:2012zj}, where the central theme is to disentangle the state in position space. 

As mentioned previously, the field theory of interest in this paper, namely the single-trace sector of the $d$-dimensional $O(N)$ vector model, is holographically dual to a higher spin theory on $AdS_{d+1}$. Indeed, there is by now a sizable literature claiming a deep relation between tensor networks and holography; see \cite{Swingle:2009bg, Evenbly:2011, Swingle:2012wq, Nozaki:2012zj, Mollabashi:2013lya, Lee:2015vla, Pastawski:2015qua, Czech:2015kbp} and references there-in. It is therefore clearly an interesting question whether we can shed light on the $AdS$/tensor network correspondence within our framework. However the spirit of the present paper is primarily field theoretic -- making contact with the holographic descriptions presented previously in \cite{Leigh:2014tza,Leigh:2014qca} will be left to future work.  Although we are not discussing real-space entanglement renormalization in the present work, we should perhaps remark in passing that for theories with higher spin duals, the holographic dictionary to compute real-space entanglement (for excited states) is not known in general dimension. Furthermore, these dual higher spin theories are not geometric in the conventional sense, and so it is unclear to what extent one can recover conventional geometries from the tensor network constructions for free theories.

The rest of the paper is organized as follows. Because we want to employ {\it exact} renormalization methods, in Section 2 we recall the construction of wave functionals in continuum free field theories. Although much of this discussion is standard material, we provide it here for completeness. In particular, we consider $N$ scalar fields and define the generator of such states by introducing sources for all $O(N)$-singlet single trace operators. As well, we give a short review of the implementation of ERG for the partition function, regarded as a functional of the operator sources. In Section 3, we discuss the implementation of ERG in the context of wave functionals and transition amplitudes. In Sections 3.1 and 3.2 the renormalization scale dependence of the ground state and excited states, respectively, is determined. The form of these equations suggest an interpretation in terms of a scale transformation and a {\it disentangling} operation (in momentum space). In Section 4, we discuss this interpretation of the solutions of the RG equations and make comparisons with Wilsonian RG and entanglement RG. We have also included a number of Appendices, which contain background material on and further details on the calculation of wave functionals in field theories, on the form of suitable regulators and further details of the exact renormalization group formalism.

\section{$O(N)$ Singlet wave functionals in Vector Models}\label{SingWFsect}
We consider a relativistic scalar field theory with $N$ scalars $\phi^a(t,\vec x)$ (where $a = 1, \cdots N$) on $d$ dimensional Minkowski spacetime ($D=d-1$ will denote the number of spatial dimensions). To construct a basis for the Hilbert space, we introduce a foliation in terms of space-like hypersurfaces, $\{\Sigma_t\}$ (where $t$ is the Minkowski time). On a particular hypersurface, $\Sigma$, we introduce a basis $|\varphi^a(\vec x)\rangle$ for the Hilbert space, with $\varphi^a(\vec x)\in L^2(\Sigma)$. 

For any generic state, it is useful to work with its overlap with the basis states $|\varphi^a(\vec x)\rangle$, that is,  the wave functional corresponding to the state.  For the theory under consideration, there is a simple and explicit construction of the wave functionals corresponding to the ground state and a class of $O(N)$-singlet excited states, in terms of a path integral with fixed boundary conditions on $\Sigma$; we will review this construction below. The action is given by
\begin{equation} \label{Fullaction}
S_\phi=\frac{1}{2}\sum_{a=1}^N\int_M dt\,d^{D}\vec x\,\phi^a(t,\vec x)\,\Box\phi^a(t,\vec x)+\frac{1}{2}\sum_{a=1}^N\int_{\Sigma}d^{D}\vec x\ \phi^a(\vec x)\,n^{\mu}\partial_{\mu}\phi^a(\vec x).
\end{equation}
where we are using the mostly plus Lorentzian signature: $\Box\equiv \partial_{\mu}\partial^{\mu}=-\partial_t^2+\vec\nabla^2$.  From here on we will drop the $O(N)$ indices and leave their sum implicit.\\
\\
From a classical point of view, the boundary term in \eqref{Fullaction} is necessary to obtain a well-defined variational principle, consistent with fixing $\phi\Big|_{\Sigma}=\varphi$, while in the path integral language, these terms enforce appropriate boundary conditions. 
Since we are in free field theory we have the luxury of performing the path integral, weighted by $S_\phi$, over field configurations that obey the boundary condition, $\phi\Big|_{\Sigma}=\varphi$.  In order to do this, it is convenient to rewrite the integration variable as a classical field satisfying the equations of motion with the boundary conditions plus quantum fluctuations that vanish on $\Sigma$:
\begin{eqnarray}
\phi=\phi_c+\chi\qquad\qquad\Box\phi_c=0\qquad\qquad\phi_c|_{\Sigma}=\varphi(\vec x)\qquad\qquad \chi|_{\Sigma}=0.
\end{eqnarray}
For completeness, this is reviewed in great detail in Appendix \ref{transamapp}.  The result is the correct ground state wave functional.  Additionally by considering the boundary of the manifold to consist of two surfaces, $\Sigma_\pm$, we show in the same appendix that the path integral weighted by $S_\phi$ produces the correct transition amplitudes.


\subsection{States}\label{sectarbstates}
As mentioned above, the procedure for constructing states from the path integral is simple. Let us first recall the construction of the ground state wave functional. Consider the \emph{Euclidean} path integral $Z[M_{-}; \varphi]$ on the lower-half space $\tau \leq 0$ (where $\tau$ is Euclidean time), with the boundary conditions $\phi(0,\vec{x}) = \varphi(\vec{x})$, and $\lim_{T \to \infty} \phi(-T,x) = 0$. From the standard time-slicing construction of this path integral,  we have
\begin{eqnarray}
Z[M_-;\varphi ] = \lim_{T \to \infty} \left\langle \varphi(\vec{x}) \left| e^{-\int_{-T}^0d\tau \hat{H}} \right| 0\right\rangle
\end{eqnarray}
where note that $|0\rangle $ is not the vacuum, but corresponds to the field configuration $\phi (\vec{x})=0$. However, by expanding $|0\rangle$ in terms of energy eigenstates, we see that the limit $T \to\infty$ isolates the vacuum state inside this expansion, and so we obtain\footnote{The projection onto the ground state pertains only when there exists a small gap, e.g., a mass term $\mu^2\phi^2$ in the action (and the state $|0\rangle$ has non-zero overlap with the vacuum).  Such a gap of course breaks the conformal invariance of the theory, so one might worry that this interferes with the renormalization group flow.  We will see later that $\mu^2$ is a special case of a bi-local source whose RG equations we will derive generically.  It will then be clear that the limit $\mu^2\rightarrow0$ is consistent with renormalization.}
\begin{eqnarray}
Z[M_-;\varphi ] \sim  \left\langle \varphi(\vec{x}) |  \Omega \right\rangle
\end{eqnarray}
where the overall normalization factor will be fixed shortly. So  we deduce that the Euclidean path integral over the lower half-space constructs the ground state wave functional $\left\langle \varphi(\vec{x}) |  \Omega \right\rangle$ (see fig 2(a)). 
\begin{figure}[!h]
\centering
\begin{tabular}{ l l l l}
\includegraphics[width=7cm]{GS.pdf} & 
\includegraphics[width=7cm]{ES.pdf}
\end{tabular}
\caption{\small{\textsf{(a) The ground state wave functional is given by the Euclidean path integral on the lower-half space. (b) Excited states can be constructed by operator insertions in the path integral.}} }
\label{fig:half}
\end{figure}


There are several directions in which we can generalize this construction. Firstly, we can construct excited states; in this paper, we will be interested in states of the form
\begin{eqnarray}\label{states}
|\psi\rangle = e^{-\delta \hat H} \hat{\mathcal{O}}_1(0,\vec{x}_1)\hat{\mathcal{O}}_2(0,\vec{x}_2)\cdots \hat{\mathcal{O}}_n(0,\vec{x}_n) |\Omega \rangle
\end{eqnarray}
where the operators $\hat{\mathcal{O}}_i$ are arbitrary, single trace $O(N)$-singlet operators (i.e., conserved currents of arbitrary spin and their descendants). In order to ensure the normalizability of these states, we will always take $\delta > 0$. These states can be constructed by performing the path integral on the lower half space, with the appropriate operator insertions at $\tau=-\delta$ (see Fig. \ref{fig:half}(b)). In fact, a convenient way to deal with these states in terms of path integrals is to package them in a \emph{generating functional}\footnote{As we will see shortly, potential divergences from the $\phi-\phi$ OPE are cancelled when we normalize these states.}
\begin{eqnarray}
|\psi [B]\rangle =  \mathcal{T}_E e^{ -\frac{1}{2}\int_{M_-} d^dx\int_{M_-}d^dy\,\hat{\phi}^a(x)B(x,y)\hat{\phi}^a(y)} |\Omega\rangle
\end{eqnarray}
where $\mathcal{T}_E$ denotes Euclidean time-ordering, and the bi-local source $B(x,y)$ can be thought of as taking the form of a ``differential operator''
\begin{eqnarray}
B(x,y) = \sum_{s=0}^{\infty} B^{(s)}_{\mu_1\cdots \mu_s}(x) \partial^{\mu_1}\cdots \partial^{\mu_s} \delta^d(x-y).
\end{eqnarray}
It should be clear from the time-slicing construction that $\mathcal{T}_E$ allows us to write the state $\psi[B]$ as the Euclidean path integral on the lower half-space, but with the action deformed by the source term
\begin{eqnarray}
\langle \varphi(\vec{x})|\psi[B]\rangle = \int^{\phi(0,\vec{x})= \varphi(\vec{x})}_{\phi(-\infty,\vec{x})\to 0} [\mathscr{D}\phi]\;e^{-S},\;\;\;S = S_\phi +   \frac{1}{2}\int_{M_-} d^dx\int_{M_-}d^dy\,\phi^a(x)B(x,y)\phi^a(y)
\end{eqnarray}
A second direction in which we can generalize is to consider the real-time evolution of the excited states discussed above, namely
\begin{eqnarray}
|\psi(t_0)\rangle = e^{-\delta \hat H -it_0\hat H} |\psi \rangle
\end{eqnarray}
This is easily accomplished by performing the path integral in complex time, along the contour $C$ shown in Fig. \ref{fig:RT.pdf} (with $t=-i\tau$ along the Euclidean section of the contour). In fact, it is a simple matter to generalize the generating functional of states to real time by extending the source $B$ along this contour -- we will refer to this generating functional as $|\psi_C[B]\rangle $. The utility of defining this generating functional is that it allows us to study the evolution of the more general class of states which are created by the contour-ordered action of operators, i.e.
\begin{eqnarray}
|\psi(t_0)\rangle_C =  e^{-\delta\hat H-it_0\hat H}\mathcal{T}_C[ \hat{\mathcal{O}}(t_1,\vec{x}_1)\cdots \hat{\mathcal{O}}(t_n, \vec{x}_n)]|\Omega \rangle .
\end{eqnarray}
The more general situation of out-of-time-order action of operators (in real time) requires a more complicated contour, and will not be considered in this paper.
\begin{figure}[!h]
\centering
\begin{tabular}{ l l l l}
\includegraphics[height=5.7cm]{RT.pdf} & \includegraphics[height=5.7cm]{norm.pdf}
\end{tabular}
\caption{\small{\textsf{(a) Operator insertions along the Euclidean branch of the contour prepares a state, while operator insertions in the real time calculate correlation functions in that state. (b) Operator and state norms are given by the path integral over the reflected contour.  Evolution by a small imaginary time $\delta$ ensures these norms do not suffer from contact divergences of operators acting at the same spacetime point.}} }\label{fig:RT.pdf}
\end{figure}

Finally, let us discuss the norm of the state $|\psi_C[B]\rangle $. The dual $\langle \psi_C[B]|$ involves a reverse of contour ordering and so the combined contour over which the path integral is performed is the time-contour $\tilde{C}$ shown in figure \ref{fig:RT.pdf}(b). Therefore, the norm is given by the partition function of the theory for the time-contour $\tilde{C}$, but where the source $B$ is taken to satisfy the reflection symmetry
\begin{eqnarray}
B(t_x,\vec{x}; t_y, \vec{y}) = B^*(t_x^*,\vec{x}; t_y^*,\vec{y})
\end{eqnarray}
For the most part, we will find it convenient to divide out by the normalization and deal with the normalized generating functional
\begin{eqnarray}
|\Psi_C[B]\rangle = \frac{1}{Z^{1/2}_{\tilde{C}}[B]}  |\psi_C[B]\rangle,\;\; Z_{\tilde{C}}[B] = \langle \psi_{C}[B] | \psi_C[B]\rangle
\end{eqnarray}
It is worth emphasizing that the state $|\Psi_C[B]\rangle$ does not generate {\it normalized} excited states directly:
\begin{eqnarray}
-i\frac{\delta}{\delta B(x,y)}|\Psi_C[B]\rangle = \Big(\hat{\cal O}(x,y) -\mathrm{Re}\,\langle \hat{\cal O}(x,y)\rangle_B\Big) |\Psi_C[B]\rangle
\end{eqnarray}
is not normalized to unity.




\subsection{Background Symmetry}\label{sectbacksymm}
Before moving on to renormalization, let us address a large background symmetry group present in our description of wave functionals. We have established that the wave functional is the path integral over field configurations with Dirichlet boundary conditions on a constant time slice $\Sigma$:
\begin{eqnarray}\label{gf1}
\langle \varphi(\vec{x})|\psi_C[B]\rangle = \int [\mathscr{D}\phi]^{\phi(t_0,\vec x)=\varphi(\vec x)}_{\phi(-\infty,\vec x)\rightarrow 0}\;e^{iS_C[\phi]},
\end{eqnarray}
where the subscript $C$ on the action indicates that we integrate along the time-contour $C$. In this path integral, $\phi$ is an integration variable and so we are always allowed a field redefinition of the form\footnote{In order to make contact with more familiar background symmetries, it is useful to take $\mathcal{L}$ to be quasi-local:
$$\mathcal{L} (x,y) = \delta^d(x-y)+\zeta^{\mu}(x)\partial_{\mu}^{(x)}\delta^d(x-y)+\zeta^{\mu\nu}(x)\partial_\mu^{(x)}\partial_\nu^{(x)}\delta^d(x-y) + \zeta^{\mu\nu\lambda}(x)\partial_{\mu}^{(x)}\partial_{\nu}^{(x)}\partial_{\lambda}^{(x)}\delta^d(x-y)+\cdots, $$
where we see that the first non-trivial term is a diffeomorphism, while the higher terms are higher spin transformations.}
\begin{equation}
\phi'(x)=\int d^{d}y\,\mathcal L(x,y)\phi(y) \equiv\left(\mathcal L\circ \phi\right)(x), \label{ol20}
\end{equation}
where we have introduced the ``$\circ$" product notation to denote an integration over interior points $y$. For now, let us ignore any subtleties associated with the boundary at $t=t_0$; we will return to these shortly.  Following \cite{Leigh:2014qca}, we require $\mathcal L$ to satisfy 
\begin{eqnarray}
(\mathcal L^T\circ \mathcal L)(x,y)=\delta^{d}(x-y), \label{ol2}
\end{eqnarray}
and further require the source to transform as
\begin{equation}\label{Btrans}
B\rightarrow \mathcal L^{-1}\circ B\circ \mathcal L
\end{equation}
Additionally, we must also introduce another source $W_{\mu}(x,y)$ to act as a background connection, which transforms as
\begin{equation}\label{Wmutrans}
W_\mu\rightarrow \mathcal L^{-1}\circ W_\mu\circ \mathcal L+\mathcal L^{-1}\circ \left\lbrack\partial_\mu,\mathcal L\right\rbrack_{\circ}.
\end{equation}
extending $\partial_\mu$ to 
\begin{equation}
D_\mu(x,y)=\partial_\mu^{(x)}\delta^{d}(x-y)+W_\mu(x,y).
\end{equation}
With these conditions, it is easy to check that the bulk action is invariant\footnote{More precisely, the transformation \eqref{ol20} induces a Ward identity that relates the partition function evaluated at one value of $B,W_\mu$ to another related to it by (\ref{Btrans},\ref{Wmutrans}).} under \eqref{ol20}.  We will refer to the group of these background symmetries of the path integral as $O(L^2)$, because these are linear transformations on square-integrable fields which satisfy the orthogonality condition \eqref{ol2}.  As was discussed in \cite{Leigh:2014qca}, it is always possible to take $W_\mu$ to be a flat connection $W_{\mu}^{(0)}$:
\begin{eqnarray}
\partial_{\mu}W^{(0)}_{\nu}-\partial_{\nu}W^{(0)}_{\mu} + \left[W^{(0)}_{\mu}, W^{(0)}_{\nu}\right]_{\circ} = 0.
\end{eqnarray}
This is because any corrections to $W^{(0)}$ under the renormalization group flow which are not flat can always be absorbed into the other source $B$. This is a special property of the bosonic theory. In this sense, the flat connection $W^{(0)}$ is associated with the free fixed point, while the source $B$ should be thought of as deforming away from the fixed point.  From here on we will assume that the background connection is flat, but for tidiness, refrain from including the $``(0)"$ superscript. 

We must now confront the fact that since we are dealing with wave functionals, our spacetime manifold has a boundary, namely the Cauchy surface $\Sigma$ at $t=t_0$.
Indeed, we have to be careful in how we treat $\mathcal L$ as we approach $\Sigma$.  For example, in a derivative expansion the first non-trivial term acts as a diffeomorphism on $\phi$. We want to restrict to diffeomorphisms which preserve the Cauchy surface; similar remarks apply to higher spin transformations.\footnote{Additionally, as we discussed above, we have tuned our sources to zero within a neighborhood of width $\delta$ around $\Sigma$. We still wish to keep this buffer zone around $\Sigma$, for normalizability.}  This means we should restrict $O(L^2)$ transformations to become bi-local only in the spatial directions within the $\delta$ neighborhood of $\Sigma$:
  \begin{equation}
  \lim_{t \to t_0} \mathcal L(t,\vec x;t',\vec x')= \ell(t,\vec x;\vec x')\delta(t-t').
  \end{equation}
We will call this subgroup $O_{\{\Sigma\}}(L^2)$ to make explicit its dependence on the constant time boundary.  This subgroup then acts on the boundary values by
  \begin{equation}
  \varphi'(\vec x)=\int d^{D}\vec x'\ell(t_0,\vec x;\vec x')\varphi(\vec x') \equiv \left(\left.\ell\right|_{t_0}\cdot\varphi\right)(\vec x).
  \end{equation}
where the ``$\cdot$" product denotes an integration along the spatial coordinates of $\Sigma$.  In allowing $O_{\{\Sigma\}}(L^2)$ to act on the boundary values of $\phi$, we should be careful to covariantize the boundary term in the action (see \eqref{Fullaction}):
\begin{align}
\int_{\Sigma}d^{D}\vec x\,\varphi \left.\partial_t\phi\right|_{t}\rightarrow&\int_{\Sigma}d^{D}\vec x\,d^{D}\vec x'\,\varphi (\vec x)\left(\partial_t\,\delta^{D}(\vec x-\vec x')+w_t(t_0,\vec x;\vec x')\right)\phi(t_0,\vec x')\nonumber\\
&\equiv \int_{\Sigma }d^{D}\vec{x}\,\varphi (t_0,\vec{x})  (D_t \cdot \phi)(t_0,\vec{x}),
\end{align}
where in the above expression we have required $W^{(0)}_\mu$ to become temporally local within the $\delta$ neighborhood of $\Sigma$:
\begin{equation} \label{c1}
\lim_{t \to t_0} W^{(0)}_\mu(t,\vec x;t',\vec x')=w_\mu(t;\vec x,\vec x')\delta(t-t'
\end{equation}

In addition to the symmetries discussed above, there is also a scaling symmetry, which is going to be relevant for the renormalization group.  Let us introduce a Weyl parameter $z$ and rescale the metric, $\eta_{\mu\nu}\rightarrow z^{-2}\eta_{\mu\nu}$.  At the same time we rescale the bi-local source $B\rightarrow z^{d+2}B$ and the connection $W_\mu\rightarrow z^d\,W_\mu$; note from \eqref{c1} that the boundary connection rescales as $w_\mu\rightarrow z^{d-1}\,w_\mu$.  The field theory action now reads
\begin{equation}
S_\phi+S_{source}=\frac{1}{2z^{d-2}}\int_M \phi(x)\circ D^2\circ \phi(x)+\frac{1}{2z^{d-2}}\int_{\Sigma}\varphi(\vec{x}) \cdot D_t\cdot\phi(\vec{x}) +\frac{1}{2z^{d-2}}\int_M \,\phi(x) \circ B\circ\phi(x).
\end{equation}
With these redefinitions, it is easy to see that 
\begin{eqnarray}
\phi \to \lambda^{\frac{d-2}{2}}\phi,\;\; z \to \lambda z
\end{eqnarray}
is a symmetry of the action.
Together with $O_{\{\Sigma\}}(L^2)$ transformations, we will sometimes denote the full group (including the above scaling symmetry) as $CO_{\{\Sigma\}}(L^2)$.  We can now write down a Ward identity for $CO_{\{\Sigma\}}(L^2)$ which succinctly encodes the background symmetries discussed above:
\begin{align} \label{WI}
\left\langle \lambda^{\frac{d-2}{2}}\ell\cdot\varphi \Big|\Psi[z,M,B]\right\rangl
&=J_{(\lambda,\ell)}^{-N/2}\Big\langle \varphi\Big | \Psi[\lambda^{-1}z,\lambda^{-1}M,\mathcal L^{-1}\circ B\circ \mathcal L]\Big\rangle
\end{align}
where by using the notation $ |\Psi[z,M,B]\rangle$, we have chosen to explicitly display the dependence of the state on the cutoff $M$ (to be introduced shortly) and the Weyl factor $z$. Additionally, $\mathcal{L}$ is an $O_{\{\Sigma\}}(L^2)$ transformation which approaches $\ell$ close to $\Sigma$, and  $J^N_{(\lambda,\ell)}$ is the Jacobian from the switching of integration variables ($\mathscr D(\lambda^{\frac{d-2}{2}}\ell\cdot\varphi)=J_{(\lambda,\ell)}^N\mathscr D\varphi$) in the normalization

\section{The Exact Renormalization Group for states}\label{regularizationsect}
Our discussion up to now has been formal; in practice, one must define a regularization scheme such that the path integrals in question exist and can be evaluated.  In this paper, following \cite{Polchinski:1983gv,Leigh:2014tza,Leigh:2014qca}, we will use a regularization scheme that eliminates the effects of the high momentum modes in the path integral by including a smooth cutoff function $K(s)$ in the action, with
\begin{equation}
K(0)=1\qquad\qquad\lim_{s\rightarrow\infty}K(s)=0.
\end{equation}
We augment our action to 
\begin{equation}\label{regaction}
S_\phi=\frac{1}{2z^{d-2}}\int_M \phi(x)\;\circ \;K^{-1}\left(-\frac{z^2}{M^2}\vec D^2\right)\circ D^2\circ \phi(x)+\frac{1}{2z^{d-2}}\int_{\Sigma}\varphi(\vec{x})\cdot K^{-1}\left(-\frac{z^2}{M^2}\vec D^2\right)\cdot D_t \cdot\left.\phi\right|_{\Sigma}(\vec{x})
\end{equation}
so that field configurations with large \emph{spatial} momentum contribute a rapidly varying phase to the path integral (or along Euclidean sections of the contour, provide exponentially suppressed contributions to the path integral). It is important to note that in doing so we are not truncating
 the Hilbert space, but we are limiting the dynamics of high momentum modes. We note that the class of cutoff functions we use in this paper is different from the cutoff functions appearing in previous works \cite{Leigh:2014tza,Leigh:2014qca} --- in particular we choose to regulate with respect to the spatial Laplacian, $\vec D^2$, as opposed to the full d'Alembertian, $D^2$.  Although we do not expect different cutoff functions to change the essential physics, there are classes of cutoff functions that are better suited for a given calculation.  A discussion of why the above cutoff is natural for studying states is given in appendix \ref{sectlorreg}, but broadly the reason is as follows: the Hilbert space at the free fixed point can be thought of as a tensor product of harmonic oscillators corresponding to each momentum mode $\vec{k}$:
\begin{eqnarray}
\mathcal{H} = \otimes_{\vec{k}\in \mathbb{R}^{d-1}} \mathcal{H}_{\vec{k}}
\end{eqnarray}
The above choice of cutoff function preserves this structure (were we to include time-derivatives inside the cutoff function, this would no longer be the case), but tunes the parameters of the high-energy oscillators ($\vec{k} \gg M/z)$ so as to exponentially suppress their dynamics. 
Furthermore, by excising time derivatives from the cutoff we can more cleanly implement a variational principle consistent with fixing field configurations at fixed times. 

Let us begin by recalling how the exact renormalization program works for the Euclidean partition function (the details of which can be found in \cite{Leigh:2014qca}).  First we start with the partition function as a functional of the bi-local source, $B$, and the source for the identity, $\mathcal U$.  This partition function is defined in the presence of a regulator with a cutoff, $M$, and a Weyl scaling parameter $z$ (defined in the previous section): $Z_E[z, M, B, \mathcal U]$.  The renormalization of the partition function is then a two step process:
\begin{itemize}
\item The first step is to lower the cutoff $M\rightarrow\lambda M$ with $\lambda<1$.  In Polchinski's exact renormalization this is done directly by changing the cutoff function; in the Wilsonian method this is analogous to integrating out the high frequency modes.  This is then interpreted as the partition function of an effective theory at lower momenta, with new values of the sources:
\begin{equation}
Z_E[z, M, B,\mathcal U]=Z_E[z, \lambda M, \tilde B, \tilde{\mathcal U}].
\end{equation}
The expressions for $\tilde B$ and $\tilde{\mathcal U}$ can be derived in detail by the methods in \cite{Polchinski:1983gv}.  
\item The next step is to perform a $CO(L^2)$ transformation, $\mathcal L$, to bring the cutoff back to $M$ while scaling the Weyl factor to $\lambda^{-1}z$.  Since, as discussed above, this is a redefinition of path integration variables, this is also an identity:
\begin{equation}
Z_E[z,\lambda M,\tilde B,\tilde{\mathcal U}]=Z_E[\lambda^{-1}z, M, \mathcal L^{-1}\circ\tilde B\circ\mathcal L, \hat{\tilde{\mathcal U}}]\equiv Z_E[\lambda^{-1}z, M, B(\lambda^{-1}z), \mathcal U(\lambda^{-1}z)],
\end{equation}
(here $\hat{\tilde{\mathcal U}}$ allows for the possibility of a $CO(L^2)$ anomaly). 
\end{itemize}

Now we are interested in adapting the above discussion to the ERG flow of states or wave functionals, as opposed to partition functions. Since we're interested in the general class of excited states \eqref{states}, it is convenient to formulate this discussion in terms of the generating functional of states $\Psi_C[B]$, as has been explained previously. To be precise, what we are interested in here is the flow in the space of states of the undeformed CFT; that is, we are thinking of the states themselves, rather than correlation functions of operators in non-trivial states. This distinction is important conceptually, but in practice the difference between sources for operators and sources for operators that occur in the definition of a state amounts to where the sources occur along the contour. Locality in time along the contour is maintained throughout. These issues are discussed further in appendices.

 How should one formulate the renormalization group equation for the generating functional of states? Since $\Psi_C[B]$ can be written as a path integral along the appropriate time contour, we expect that the renormalization can be carried out by following the same steps as in the case of the Euclidean partition function; indeed, this is what we will do. The corresponding path integral will give us a one-parameter $\Psi_C[z,B]$, which we treat as the \emph{Wilsonian effective generating functional} of states. 
Thus, ERG for states can again be stated as a two-step process:

\textbf{Step 1: Lower the cutoff}
\\\\
To be explicit, recalling eq. \eqref{gf1}, the generating functional of states is given by the path integral
\begin{equation}
\Psi_C[z,M, B,\varphi]=\left\langle \varphi | \Psi_C \right\rangle = \mathcal N\int\left[\mathscr D\phi\right]^\varphi\exp\left(iS_\phi+iS_{source}\right)
\end{equation}
where $S_\phi$ is defined as in eq. (\ref{regaction}).
From here we can compute $M\frac{\partial}{\partial M}\Psi$ from the standard Polchinski formalism, the only subtlety arising from treating the boundary terms carefully.  This is done in detail in appendix \ref{ERGdetapp} and we will simply quote the result here:
\begin{align}\label{lowercutoff} 
M&\frac{\partial}{\partial M}\Psi[z,M,B,\mathcal U;\varphi]\nonumber\\
&=\left(-z\text{Tr}_{\Sigma\times C}\left(B\circ\Delta_B\circ B\circ\frac{\delta}{\delta B}\right)-z\frac{N}{4}\text{Tr}_\Sigma\left(\Delta_\Sigma\right)-\frac{z}{2}\int_\Sigma\varphi \cdot \Delta_\Sigma\cdot\frac{\delta}{\delta\varphi}\right)\Psi[z,M, B, \mathcal U; \varphi].
\end{align}
In this formula we have defined 
\begin{eqnarray}
\Delta_B=\left(D^2\right)^{-1}\circ \frac{M}{z}\frac{d}{dM}K,
\end{eqnarray}
namely, the derivative of the field theory two-point function with respect to $M$, and
\begin{eqnarray}
\Delta_\Sigma= \left. K^{-1}\frac{M}{z}\frac{d}{dM}K\right|_\Sigma
\end{eqnarray}
is a boundary kernel.  Note that, since $K$ involves a flat connection, there is no ordering ambiguity in these definitions.\footnote{We note that a term of the form $\text{Tr}_{\Sigma\times C}\left(\Delta_B\circ B\right)$ that was present in previous ERG calculations \cite{Leigh:2014tza,Leigh:2014qca}, is cancelled due to the normalization of the wave functional.  The normalization is additionally responsible for the appearance of the $\text{Tr}_\Sigma(\Delta_\Sigma)$ term which arises from an integration by parts inside of the $\int[\mathscr D\varphi]$ integral.}

\textbf{Step 2: Scale transformation}\\\\
The second step is now to raise the cutoff by performing an infinitesimal $CO_{\{\Sigma\}}(L^2)$ transformation with $\lambda=1-\varepsilon$.  We parameterize this transformation by $\mathcal L=1+\varepsilon z\,W_z+O(\varepsilon^2)$ in the interior and $\ell =1+\varepsilon z\,w_z+O(\varepsilon^2)$ on $\Sigma$.
Using the Ward identity \eqref{WI}, we then obtain
\begin{empheq}[box=\widefbox]{align}\label{totalzexcstate}
z\frac{\partial}{\partial z}\Psi=\left(z\text{Tr}_{\Sigma\times C}\left(\left([W_z,B]_\circ+B\circ \Delta_B\circ B\right)\circ\frac{\delta}{\delta B}\right)+z\frac{N}{2}\text{Tr}_\Sigma g+z\int_\Sigma\varphi\cdot g^t \cdot\frac{\delta}{\delta\varphi}\right)\Psi.
\end{empheq}
where we have defined the bi-local kernel 
\begin{equation} \label{g}
g(z;\vec x,\vec y):=\left(\frac{1}{2}\Delta_\Sigma+w_z\right)(\vec x,\vec y)
\end{equation}
The trace over $\Sigma$ of $w_z$ arises from the Jacobian of the infinitesimal $CO_{\{\Sigma\}}(L^2)$ transformation. From the above equation we see that $\mathcal U$ plays only a spectator role in the ERG equation; from here on, we will drop it from the notation. We now want to understand the meaning of the various terms appearing in the above equation. We remark in passing that whereas in the case of the partition function, the RG principle is often stated as the partition function being independent of the value of the cutoff $M$, in the case of states, we simply organize the calculation in such a way as to eliminate derivatives with respect to $M$. Later we will see that this leads to an interpretation in which the ERG acts unitarily on states.

\subsection{Ground state}
Let us first investigate the above equation for the ground state wave functional.  This means we turn all sources off along the Euclidean section of our contour.  As we can see from \eqref{totalzexcstate}, this is consistent: setting $B=0$ completely eliminates the $\frac{\delta}{\delta B}$ terms in the differential equation and no extra operators are pulled down as we move along $z$:
\begin{equation}\label{pazGS}
z\frac{\partial}{\partial z}\Psi_\Omega[z,M,\varphi]=\left(z\frac{N}{2}\text{Tr}_\Sigma\left(\frac{1}{2}\Delta_\Sigma+w_z\right)+z\int_\Sigma\varphi\cdot\left(\frac{1}{2} \Delta_\Sigma+w_z^{\text t}\right)\cdot\frac{\delta}{\delta\varphi}\right)\Psi_\Omega[z,M,\varphi].
\end{equation}
That is, the ERG flow induces no mixing of the ground state with other states at the UV fixed point.
Let us rearrange this equation slightly, to make it somewhat more transparent.  Recall that in the wave functional representation of a state, $\varphi$ and $\frac{\delta}{\delta\varphi}$ are the operators $\hat\phi$ and $\hat\pi$, respectively, acting on $\Psi_\Omega$:
\begin{equation}
\langle\varphi|\hat\phi(\vec x)|\Omega\rangle=\varphi(\vec x)\Psi_\Omega[\varphi]\qquad\qquad\qquad\langle\varphi|\hat\pi(\vec x)|\Omega\rangle=-i\frac{\delta}{\delta\varphi(\vec x)}\Psi_\Omega[\varphi].
\end{equation}
Together with \eqref{pazGS}, this then implies that the ground state satisfies the following flow equation in the $z$ direction
\begin{eqnarray} \label{quasidisent}
z\frac{d}{d z}\Big|\Omega(z)\Big\rangle &= & i\Big(\boldsymbol{K}(z)+\boldsymbol{L}(z)\Big) \Big|\Omega(z)\Big\rangle,\nonumber\\
\boldsymbol{K}(z) &=&  \frac{z}{2}\left(\hat\pi\cdot \Delta_{\Sigma} (z)\cdot\hat\phi+\hat \phi\cdot \Delta_{\Sigma}^{\text t}(z)\cdot\hat \pi\right)\nonumber\\
\boldsymbol{L}(z) &=&  \frac{z}{2}\left(\hat\pi\cdot w_z(z)\cdot\hat\phi+\hat \phi\cdot w_z^{\text t}(z)\cdot\hat \pi\right)
\end{eqnarray}
Both $\boldsymbol{K}$ and $\boldsymbol{L}$ are Hermitian operators. The operator $\boldsymbol{K}(z)$ essentially damps out the dynamics of the UV modes, while preserving the direct product structure (in momentum space) of the vacuum. Of course, as we will see in the following section, generic excited states are not product states in momentum space -- in this case then, the operator $\boldsymbol{K}(z)$ will play the role of a momentum space \emph{disentangler}, and will remove entanglement between modes above and below the scale $M/z$. Furthermore, by a judicious choice of the cutoff function (such as if we take it to be the exponential function) the operator $\boldsymbol{K}$ may be thought of as a local operator in position space. On the other hand, the quasi-local operator $\boldsymbol{L}$ implements the scale transformation which appears in the second step of ERG, in addition to a possible $O_{\{\Sigma\}}(L^2)$ transformation.

Interestingly, the differential equation \eqref{quasidisent} can be formally solved in terms of the path-ordered exponential of $\boldsymbol{K}+\boldsymbol{L}$:
\begin{equation}\label{GSpathordexp}
\Psi_\Omega[z_*,\varphi]=\Big\langle\varphi\Big|\Omega(z_*)\Big\rangle=\Big\langle\varphi\Big|\mathcal Pe^{\frac{i}{2}\int_\epsilon^{z_*}dz\int_\Sigma\left(\hat\pi\cdot g(z)\cdot\hat\phi+\hat\phi\cdot g^{\text t}(z)\cdot\hat\pi\right)}\Big|\Omega(\epsilon)\Big\rangle.
\end{equation}
where $|\Omega(\epsilon)\rangle$ is the UV ground state, and the kernel $g(z)$ was defined in \eqref{g}.  It is illuminating to write down this wave functional explicitly in the case where we choose to be a simple scale transformation at each $z$, $w_z(\vec x,\vec y)=-\frac{1}{z}\frac{(d-2)}{2}\delta^{D}(\vec x-\vec y)$.  Then solving \eqref{pazGS}, we obtain
\begin{eqnarray}\Psi_\Omega[z,M,\varphi]={\det}_\Sigma\left(z^{2-d}K^{-1}|\vec D|\right)^{N/4}\exp\left(-\frac{1}{2z^{d-2}}\int\frac{d^{D}\vec k}{(2\pi)^{D}}\,\varphi(-\vec k)K^{-1}\left(\frac{z^2}{M^2}\omega_{\vec k}\right)\omega_{\vec k}\,\varphi(\vec k)\right)
\end{eqnarray}
with $\omega_{\vec k}$ satisfying
\begin{equation}
\left(\left.\vec D^2\right|_\Sigma\cdot\psi_{\vec k}\right)(\vec x)=-\omega_{\vec k}^2\psi_{\vec k}(\vec x).
\end{equation}
for some complete set of eigenfunctions\footnote{e.g. in the gauge $w_\mu=0$, they can be chosen to be the plane waves $\psi_{\vec k}(\vec x)\sim e^{i\vec k\cdot\vec x}$ and $\omega_{\vec k}=\sqrt{\vec k^2}$.} $\psi_{\vec k}$ and $\varphi(\vec x)=\int\frac{d^{D}\vec k}{(2\pi)^{D}}\varphi(\vec k)\psi_{\vec k}(\vec x)$.  Similarly the determinant prefactor is defined by the product of $\omega_{\vec k}$.  Reassuringly, one can check that this wave functional defines the ground state at a scale $z$ by either explicitly doing the path integral defining it, or by canonical means, both of which are done in appendix \ref{transamapp}.

\subsection{Excited states}

Now let us investigate the flow of excited states, i.e. keeping $B\neq 0$, and make some comments about the nature of the RG equation, \eqref{totalzexcstate}.  We can rewrite this equation in terms of the newly defined Hermitian operators $\boldsymbol{K}$ and $\boldsymbol{L}$ as
\begin{eqnarray}\label{statedepop}
z\partial_z \left|\Psi_C[B]\right\rangle =\left(-\text{Tr}\,\boldsymbol{\beta}\circ\frac{\delta}{\delta B}+ i\boldsymbol{K} + i\boldsymbol{L}\right) \left| \Psi_C[B]\right\rangle
\end{eqnarray}
where we have defined the bi-local beta function
\begin{eqnarray}
\boldsymbol{\beta}(x,y) = -z\Big([W_z,B]_\circ +B\circ \Delta_B\circ B\Big)(x,y)
\end{eqnarray}
However, we note that while the $\boldsymbol{K}+\boldsymbol{L}$ term in \eqref{statedepop} is a Hermitian operator defined without any explicit reference to the state on which it acts, the beta function term depends explicitly on the source $B$ preparing the state.  Since $|\Psi_C[B]\rangle$ is the generating functional for excited states, it should be clear from equation \eqref{statedepop} that the beta function term induces a mixing between the states $|\Omega\rangle,\, \mathcal{O}|\Omega\rangle,\; \mathcal{O} \mathcal{O} |\Omega\rangle, \ldots$ as we move in $z$.

A natural way to think of how RG is changing the initial state is to think of the generating functional of states $\Psi_C[B]$ as being a family of states parametrized by the source, $B$.   
In this language, we have a one-dimensional vector space (spanned by $\Psi_C$) fibered over the (infinite dimensional) manifold, $\mathcal M_B$, coordinatized by the bi-local sources $B(x,y)$. This base space is a generalization of the usual notion in RG of the space of couplings, here appearing as a parameterization of states. The beta function term in the RG equation for $\Psi_C$ can then be interpreted in terms of a flow along the vector field $\boldsymbol{\beta}$ on $\mathcal{M}_B$. In fact, if we interpret $z$ as parameterizing a curve\footnote{Note that we are using the notation ${\cal B}(z)$ to denote the analogue of a `running coupling', but we re-emphasize that here the sources are regarded as encoding non-trivial states of the CFT, rather than corresponding to turning on non-trivial couplings in the CFT.} 
\begin{eqnarray}
\mathcal B: \mathbb R^+\rightarrow\mathcal M_B
\end{eqnarray}
then $\Psi_C[z,\mathcal B(z)]$ flows along this particular curve\footnote{In this case the above equation admits another interesting interpretation -- as described previously, the state $\Psi_C[B]$ defines a line bundle over $\mathcal{M}_B$, equipped with a \emph{Berry connection} $\mathcal{A}$. In this context, the $\partial_z\mathcal B(z)$ term plays the role of the Berry connection pulled back to the RG curve through the space of sources.} via
\begin{equation}
\frac{d}{dz}|\Psi_C[z,\mathcal B(z)]\rangle=\frac{\partial}{\partial z}|\Psi_C[z,\mathcal B(z)]\rangle+\text{Tr}\,\left(\partial_z \mathcal B(z)\circ\frac{\delta}{\delta \mathcal B(z)}\right)|\Psi_C[z,\mathcal B(z)]\rangle.
\end{equation}
We see from \eqref{statedepop} that along special curves solving 
\begin{eqnarray}
z\partial_z\mathcal B=\boldsymbol{\beta}(\mathcal{B}), \label{brg}
\end{eqnarray}
namely the integral curves of the vector field $\boldsymbol{\beta}$, the state $\Psi_C[z,\mathcal B(z)]$ has a particularly simple, \emph{unitary} flow equation:
\begin{equation}\label{excstpartrans}
z\frac{d}{dz}\left | \Psi_C[z,\mathcal B(z)]\right\rangle =
i\left(\boldsymbol{K} + \boldsymbol{L}\right) \left|\Psi_C[z, \mathcal{B}(z)] \right\rangle 
\end{equation}
of the same form as \eqref{quasidisent}.
\begin{figure}[H]
\centering
\includegraphics[width=.5\textwidth]{sourcemanifold-eps-converted-to.pdf}
\caption{Along integral curves $\mathcal B(z)$ of the beta function, $|\Psi[z;\mathcal B(z)]\rangle$ undergoes a unitary flow generated locally by the Hermitian operator $\mathcal{K} = \boldsymbol{K} + \boldsymbol{L}$.}
\label{sourcemanifoldfig}
\end{figure}
We note that in the picture we have presented, $\mathcal B$ prepares a state for the Hilbert space on $\Sigma$.  However, if we were to regard $\mathcal B(z)$ instead as a coupling for single-trace operators modifying the action of the theory and then $\Psi_C$ as the corresponding ground state, then \eqref{brg} is simply the ERG equation for the coupling $\mathcal{B}$.  In this case the flow of the ground state is captured entirely by the action of the Hermitian operators $\boldsymbol{K}$ and $\boldsymbol{L}$ {\em sans} any mixing with other states.  This suggests that taking the state to flow along integral curves of $\boldsymbol{\beta}$ is a natural requirement for ERG.  We argue that, in fact, unitary evolution of the state under ERG should be taken as an \emph{RG principle}.  In particular, the inner product of two unnormalized states is the Euclidean partition function.  Requiring that inner products remain unchanged through the flow in $z$ is then equivalent to requiring partition functions to be ERG invariants.   

We end this section by remarking that it is the structure of the ERG setup that allows us to require a unitary RG principle.  In particular the regulation of the theory is implemented not by excluding the high energy modes from the Hilbert space, but instead by altering the action (or equivalently the Hamiltonian) of the theory.  Because of this, lowering the cutoff does not change the size of the Hilbert space; the modes above the cutoff remain ancillary.  This is analogous to the ``exact holographic map" variant of entanglement renormalization \cite{Lee:2015vla}.  In the next section we will see that $\mathcal K(z)=\boldsymbol{K}+\boldsymbol{L}$ has a natural home in the language of unitary networks and we will expound upon the unitary structure of the ERG.

\section{The ERG as a tensor network}\label{cMERAsect}

We now want to better understand the right hand side of equation \eqref{statedepop}, and in particular the meaning of the hermitian operations appearing there.  To facilitate this, we will focus on the one parameter family of excited states, $\left|\Psi(z)\right\rangle \equiv \Big|\Psi[z,\mathcal B(z)]\Big\rangle$, evolving along the integral curve $z\partial_z\mathcal B=\boldsymbol{\beta}$ and which satisfies \eqref{excstpartrans}, or equivalently:
\begin{equation} \label{tn}
\left|\Psi(z)\right\rangle=\mathcal{P}\,e^{i \int_{z_{IR}}^{z}\frac{du}{u}\, \mathcal{K}(u)}\left|\Psi(z_{IR})\right\rangle,
\end{equation}
where $\mathcal{K} = \boldsymbol{K}+\boldsymbol{L}$. The scale-dependent unitary evolution under RG which we have found is not a novel concept.  This is in fact a central feature in (multi-scale) entanglement renormalization, MERA, which is a particular implementation of a \emph{tensor network} representation of certain special states (namely the ground state and low-energy excited states of a critical system). In MERA, a discrete system of finite size is ``evolved" step-by-step under the action of \emph{local} (where by local we mean action on nearest-neighbour sites) unitary\footnote{More precisely, these are taken to be isometries, but by adding ancillary degrees of freedom, it is possible to think of them as unitary operators.} operators, thus building a web of unitary operations culminating in an infrared state. The central precept of MERA is to choose the local unitaries carefully so as to spatially disentangle the state at each scale, thus leading to an IR state which is completely disentangled spatially. Continuous versions of MERA (cMERA) have also been proposed and studied \cite{Haegeman:2011uy,Nozaki:2012zj} for free quantum field theories, and are designed such that many of the above features carry over.

In our case, we see from equation \eqref{tn} that ERG naturally gives us a quasi-local continuum unitary evolution, reminiscent of cMERA. 
So what is the operator $\mathcal{K}$ doing at each step of ERG?  From its definition we see that it is the combination of a scale transformation $\boldsymbol{L}$ and the operator $\boldsymbol{K}$ which freezes the UV modes of the state above the given scale.  This is of course remarkably similar to the hermitian operation which is taken to generate the cMERA in free field theories -- the combined action of a scale transformation and a \emph{disentangler} which removes real-space entanglement at each RG step. So then the question which remains to be answered is whether we can interpret the operator $\boldsymbol{K}$ which appears in ERG as a disentangler -- we claim that in fact $\boldsymbol{K}$ plays the role of a \emph{disentangler in momentum space.} To illustrate this, we will consider a momentum-space reduced density matrix and track how it changes as momentum modes are traced over.

Before proceeding with this,  let us clarify the differences between the structure of the ERG setup and the familiar Wilsonian RG. In the latter, there is a hard cutoff in momentum space, and the RG is obtained by lowering the cutoff by explicitly integrating over the degrees of freedom within shells in momentum space. In Ref. \cite{Balasubramanian:2011wt}, it was shown that this leads to a reduced density matrix which is mixed. In the ERG setup on the other hand, all modes are present in the path integral but the high momentum structure of the Hamiltonian of the theory is modified by the presence of the cutoff function, and ERG corresponds to lowering the scale of the cutoff function. Nevertheless, we can still introduce the concept of a reduced density matrix in momentum space in the ERG setup, by explicitly tracing over modes with momenta above a given scale $\mu$. Such a process then can be expected to give rise to a path integral that is formally similar to that employed in Wilsonian RG. 

So we will now consider the momentum-space reduced density matrix of a specific excited state.  (Of course the ground state of the system is a product state in momentum space and so the calculation is trivial in that case; in order to understand the effect of $\boldsymbol{K}$ we need to look at an excited state.)  For simplicity, let us pick the state created  by acting on the vacuum by a singlet operator which is local in position space\footnote{Since $\Big|\Psi_{\phi\phi}\Big\rangle$ involves a momentum integration, it is not a product state in momentum space.}:
\begin{equation}
\Big|\Psi_{\phi\phi}(z)\Big\rangle=\mathcal N\hat\phi(\vec x)\hat\phi(\vec x)\Big|\Omega(z)\Big\rangle.
\end{equation}
The matrix elements of the density matrix in the  $\varphi(\vec{p})$ basis are then:
\begin{equation}
\Big\langle\varphi_1(\vec p)\Big|\rho_{\phi\phi}\Big|\varphi_2(\vec q)\Big\rangle=\left|\mathcal N\right|^2\int_{\vec p_{1,2},\vec q_{1,2}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!e^{i\sum(\vec p+\vec q)\cdot\vec x}\varphi_1(\vec p_1)\varphi_1(\vec p_2)\varphi_2(\vec q_1)\varphi_2(\vec q_2)e^{-\frac{1}{2z^{d-2}}\int_{\vec k}\left(\varphi_1(-\vec k)K^{-1}\omega_{\vec k}\varphi_1(\vec k)+\varphi_2(-\vec k)K^{-1}\omega_{\vec k}\varphi_2(\vec k)\right)}
\end{equation} 
Now we choose some reference momentum $\mu$, and calculate the reduced density matrix by tracing out all of the degrees of freedom with momenta $|\vec k|\geq\mu$, that is
\begin{equation}
\Big\langle\varphi_{1,<}(\vec p)\Big|\rho_{\phi\phi,<}\Big|\varphi_{2,<}(\vec q)\Big\rangle:=\int\prod_{|\vec k|\geq\mu}d\tilde\varphi_{>}(\vec k)\Big\langle\varphi_{1,<}(\vec p)\Big|\otimes\Big\langle\tilde\varphi_>(\vec k)\Big|\rho_{\phi\phi}\Big|\varphi_{2,<}(\vec q)\Big\rangle\otimes\Big|\tilde\varphi_>(\vec k)\Big\rangle.
\end{equation}
The Gaussian weight factorizes nicely under the splitting of the Hilbert space and the two-point and four-point integrations of $\tilde\varphi_>(\vec k)$ are performed easily.  Up to overall normalization we obtain
\begin{align}\label{reddenmat}
\rho_{\phi\phi,<}\sim& \left\{(N^2+2N)z^{2(d-2)}\int_{|\vec p|\geq\mu}\frac{K(\vec p)}{\omega_{\vec p}}\int_{|\vec q|\geq\mu}\frac{K(\vec q)}{\omega_{\vec q}}\right.\nonumber\\
&\left.\qquad+z^{d-2}\int_{|\vec q|\geq\mu}\frac{K(\vec q)}{\omega_{\vec q}}\int_{|\vec p_{1,2}|<\mu}\!\!\!\!\Big(N\varphi_{1,<}(\vec p_1)\varphi_{1,<}(\vec p_2)+N\varphi_{2,<}(\vec p_1)\varphi_{2,<}(\vec p_2)+4\varphi_{1,<}(\vec p_1)\varphi_{2,<}(\vec p_2)\Big)e^{i\sum\vec p\cdot\vec x}\right.\nonumber\\
&\qquad+\left.\int_{|\vec p_{1,2}|,|\vec q_{1,2}|<\mu}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!e^{i\sum(\vec p+\vec q)\cdot\vec x}\varphi_{1,<}(\vec p_1)\varphi_{1,<}(\vec p_2)\varphi_{2,<}(\vec q_1)\varphi_{2,<}(\vec q_2)\right\}e^{-\frac{1}{2z^{d-2}}\int_{|\vec k|<\mu}\left(\varphi_1K^{-1}\omega_{\vec k}\varphi_1+\varphi_2K^{-1}\omega_{\vec k}\varphi_2)\right)}
\end{align}
We see that the third line of \eqref{reddenmat} has the form of the original density matrix, restricted to the subspace of momenta less than $\mu$.  If the terms in the first and second lines were absent, then the reduced density matrix would be pure and identical in form to the original density matrix.  However, the presence of these terms indicates that the reduced density matrix is mixed and therefore this state would have non-zero entanglement entropy in momentum space.  Note though that each of these terms is weighted by factors of the cutoff function.
Thus, for scales $\mu^2>>\frac{M^2}{z^2}$, these integrals give vanishing contributions.  That is at scales greater than the cutoff, the reduced density matrix is essentially pure and the momentum space entanglement entropy at this scale vanishes.  Because the tracing out of $\tilde\varphi_>$ can be thought of in terms of  Wick contractions, this argument generalizes nicely to states formed by any polynomial of $\hat\phi\hat\phi$; higher order polynomials only give increasing powers of $K$.  
Because of the form of the cutoff function, the states with momenta much larger than the cutoff are disentangled from those with momenta less than the cutoff. In the ERG calculation, we note that the kernel $\Delta_\Sigma$ involved in the definition \eqref{quasidisent} of $\boldsymbol{K}$ is determined by the cutoff function alone, and is peaked at the renormalization scale. Consequently, the operator $\boldsymbol{K}$ implements the disentangling of the modes at momentum $\frac{M}{z}$ from low-energy modes. 

In terms of quantum information theory, the action of the unitary operator $\mathcal{P}\,e^{i \int_{z_{IR}}^{z}\frac{du}{u}\, \mathcal{K}(u)}$ is somewhat analogous to \emph{compression}; it packages the information in momentum space to modes below the effective cutoff scale $\frac{M}{z}$, with the density matrix above this scale being in a trivial product state. From this point of view, these UV modes are analogous to ancillary degrees of freedom required to implement unitary evolution in MERA. As we take $z$ larger, the subsystem carrying the entanglement gets increasingly smaller, but throughout the process the size of the Hilbert space remains the same and since the action is unitary, no information is lost.  

\begin{figure}[H]
\centering
\includegraphics[width=.5\textwidth]{quantumcircuit-eps-converted-to.pdf}
\caption{The darker region indicates entanglement in momentum space.  The disentangler acts as a unitary channel that pushes this to lower and lower scales.}
\label{figcircuit}
\end{figure}

The arguments above only rely on the asymptotic behavior of $K$ at large momenta.  Of course, these asymptotics are fixed by requiring that loop integrals are UV finite via modifying the propagator:
\begin{equation}
\frac{i}{p^2+i\epsilon}\rightarrow \frac{iK\left(\frac{z^2\vec p^2}{M^2}\right)}{p^2+i\epsilon}.
\end{equation}  
That is, convergence of the field theory $n$-point functions naturally \emph{fixes} that the IR state has no momentum space entanglement. This is simply the organization given to us by ERG, given that there is a local (UV) free fixed point theory. 
We contrast this property of the ERG tensor network with the tensor network of cMERA.  In cMERA, a particular \emph{spatially} disentangled IR state is \emph{chosen}; the appropriate unitary transformations that connect it to the UV entangled state are determined by a variational principle. 
Now let us suppose that we could build the cMERA tensor network from the ERG process. As we saw above, the disentangler\,is determined by a choice of cutoff function, so let us suppose that there existed a cutoff function that interpolates between the UV ground state at $z=\varepsilon<<1$ and the spatially unentangled IR ground state of \cite{Nozaki:2012zj}:
\begin{displaymath}
\xymatrix{
\Psi_\Omega[z,\varphi]=\mathcal N\exp\left(-\frac{1}{2z^{d-2}}\int_\Sigma\frac{d^{D}\vec k}{(2\pi)^{D}}\varphi(-\vec k)K^{-1}\left(\frac{z^2\omega_{\vec k}^2}{M^2}\right)\omega_{\vec k}\varphi(\vec k)\right)\ar[d]^{z\rightarrow\varepsilon<<1} \ar[dr]^{z\rightarrow\infty}&\\
\Psi_{UV}[\varphi]=\mathcal N_{UV}\exp\left(-\frac{1}{2\varepsilon^{d-2}}\int_\Sigma\frac{d^{D}\vec k}{(2\pi)^{D}}\varphi(-\vec k)\omega_{\vec k}\varphi(\vec k)\right)&\!\!\!\!\!\!\!\!\Psi_{IR}[\varphi]=\mathcal N_{IR}\exp\left(-\frac{1}{2z^{d-2}}\int_\Sigma\,\varphi(\vec x)\frac{M}{z}\varphi(\vec x)\right)}.
\end{displaymath}
It is easy to see that $\Psi_{IR}$ defines a state with no {\it spatial} entanglement: it is a product state in position space.  One possible cutoff function that would implement this could be taken to be
\begin{equation}
K_{cMERA}^{-1}\left(\frac{z^2\omega_{\vec k}^2}{M^2}\right)=\exp\left(\frac{1}{2}\log\left(\frac{M^2}{z^2\omega_{\vec k}^2}\right)\Theta\left(1-\frac{z^2\omega_{\vec k}^2}{M^2}\right)\right)=\left\{\begin{array}{cc}1&\qquad\frac{z^2\omega_{\vec k}^2}{M^2}<1\\
\frac{M}{z\omega_{\vec k}}&\qquad\frac{z^2\omega_{\vec k}^2}{M^2}\geq1\end{array}\right.
\end{equation}
Choosing such a cutoff function would then bridge the gap between the ERG and cMERA tensor networks.  More generally, requiring an ERG flow to the $\Psi_{IR}$ ansatz in the cMERA network requires the asymptotic behavior
 \begin{equation}
K_{cMERA}(s)=\left\{\begin{array}{cc}1&\qquad s<<1\\\sqrt s&\qquad s>> 1\end{array}\right.
\end{equation}
This does the opposite of what a regulator in a QFT is supposed to do --- it is enhancing the loop integrals in the large momentum regime.  The ``renormalization" of cMERA is then in this sense a qualitatively different procedure than renormalization in the Wilsonian or ERG sense.  

\section{Discussion}

In this paper we have applied the exact renormalization group procedure to path integral representations of states directly in continuum field theory.  In the process we have discovered that ERG naturally organizes itself into a continuous tensor network.  The nature of this tensor network is encoded in the Polchinski cutoff function and its purpose is to move entanglement in momentum space into an increasingly smaller domain of Hilbert space set by an effective renormalization scale, $M/z$.  Despite having a wide range of cutoff functions that one can use in ERG, the behavior of this network is set by the asymptotics of the cutoff function.  Ensuring that the correlators in the field theory are UV finite implies unambiguously that the IR state is one of vanishing momentum space entanglement.\\
\\
This should be contrasted with a very similar renormalization prescription for ground states of critical systems, cMERA.  In those prescriptions the IR state is chosen to be spatially unentangled and the tensors comprising the network are set by a variational principle.  We have argued that such renormalization prescriptions, while perfectly natural to consider in the broader context of unitary quantum circuits, are fundamentally different than renormalization in the Wilsonian sense.  They do not regulate UV divergences appearing in loop integrals. 
However, we note that in our discussion in the last section, it is the part of the cutoff function {\it above} the renormalization scale that differs between ERG and cMERA. Indeed, there is no doubt that the discrete version, MERA, where a fixed cutoff is provided by the lattice spacing, reproduces Wilsonian RG flow, which can be seen for example, in real space entanglement \cite{Evenbly:2008hca}.

In the recent literature, there has been a growing interest in the holographic interpretations of MERA \cite{Miyaji:2015fia, Lee:2015vla, Mollabashi:2013lya, Nozaki:2012zj} and it is clear that such schemes are capturing key features of holographic duality, particularly the nature of holographic entanglement entropy.  Early explanations of this involved interpreting the MERA network as living on and describing a discrete geometry of a constant-time slice of AdS \cite{Swingle:2009bg, Nozaki:2012zj}.  However, there is growing evidence that the connection between MERA and AdS geometry is not so direct.  For example,  the MERA tensor network fails consistency conditions for resolving length scales below the AdS radius \cite{Bao:2015uaa}.  Furthermore several groups have suggested a connection between MERA and de Sitter geometry on kinematic space
\cite{Czech:2015qta, Czech:2015kbp, Czech:2016xec}. 
In this paper, instead of extrapolating holographic geometry from a tensor network we have started with a system whose holographic geometry is well understood and extracted a tensor network from it.  Despite appearing superficially similar, this network has some important differences from cMERA.

There are several open questions at this stage.  The first is how do multi-trace interactions change the tensor network story, if at all?  It is well known that double trace deformations in the field theory amount to a change of boundary conditions in the bulk of AdS at large $N$, and the ERG framework can be extended to account for generic multi-trace interactions \cite{LP}.  It would be interesting to extend this program further to the context of the renormalization of non-trivial states.  

In this paper, we have laid the groundwork for the study of the renormalization of wave functionals in field theories. 
A natural extension would be to construct explicit reduced density matrices and hence to study the RG evolution of real space entanglement. 
Lastly, and perhaps most intriguingly, is the connection between ERG and quantum information.  
There have been many recent proposals for how quantum information encodes the key features of holographic duality from bulk reconstruction via quantum error correcting codes \cite{Almheiri:2014lwa, Pastawski:2015qua, Verlinde:2012cy}, threads of quantum information encoding the Ryu-Takayanagi formula for entanglement entropy \cite{Freedman:2016zud}, and the role of chaos and complexity in holographic geometry \cite{Stanford:2014jda, Roberts:2014ifa, Shenker:2013pqa, Shenker:2014cwa, Maldacena:2015waa}.  It is clear that there is a vast informational structure encoded in the holographic correspondence.  The tensor network constructed from ERG affords us only a glimpse at this structure.  Future work will explore and attempt to make this connection more explicit.

\vskip .5cm

{\bf Acknowledgements:}
Research supported in part by the U.S. Department of Energy contract DE-FG02-13ER42001.



