Major Project (Grant No. 2019SHZDZX01). YW is grateful for the Hospitality of
the Perimeter Institute during his visit, where the main part of this work is
done. We thank Martin Bojowald for discussions and useful suggestions.

##  Appendix A Poisson bracket for fluctuations of arbitrary order

Here, give a prescription for finding arbitrary but specified fluctuations
$\Delta^{(n)}$. The prescription is most useful for brackets between
$\Delta^{(n)}$ that are at least both of cubic order or higher. (Quadratic-
cubic brackets and quadratic-quadratic brackets can be easily “read out” using
the Leibniz rule and intuition from classical Poisson brackets.) The key
technique used here is the Baker-Campbell-Haussdorf (BCH) formula. We will
therefore call the prescription derived in this section the BCH-Poisson
bracket prescription — or BHC-PB prescription for short.

A naive adaptation of the quantum mechanical derivation in [1] — by turning
the weighted sums of canonical variables into weighted integrals — does not
work well in practice. This is because, with this naive implementation, one
obtains integrated versions of Poisson brackets over momenta, as opposed to
the form of (3.11). There is no good way to get rid of the integrals,
especially when there are constant terms like the last line of the last
equation in (3.11).

So, we will use an alternative strategy to obtain Poisson brackets without
integrations over momenta. The only price we pay is that we have to repeat the
prescription for each distinct bracket. Fortunately, once the Poisson brackets
are derived for a given order, they can be used for any physical system up to
that order. Additionally, there are not that many of them. For example, for
any system with arbitrary cubic interactions, there are only eight non-trivial
cubic-cubic Poisson brackets. (For $\varphi^{3}$-interactions, only three of
these cubic-cubic brackets are needed.)

###  A.1 The derivation

Let us now proceed with the derivation. For simplicity of demonstration, we
derive the prescription for the
$\varphi_{1}\pi_{2}\pi_{3}-\pi_{a}\varphi_{b}\varphi_{c}$ bracket in $1+1$
dimensions 101010The results of this section can be generalized to $3+1$
dimensions by replacing all $(2\pi)\delta(\dots)$ with
$(2\pi)^{3}\delta^{3}(\dots)$.

$\displaystyle\\{\Delta(\varphi_{1}\pi_{2}\pi_{3}),\Delta(\pi_{a}\varphi_{b}\varphi_{c})\\}$

and then apply the prescription in a few examples. (The subscripts
$1,2,3,a,b,c$ label the momenta $k_{1},k_{2},k_{3},k_{a},k_{b},k_{c}$.)

To utilize the BCH formula, we further introduce the following notation

$\displaystyle A(\alpha)=$ $\displaystyle\langle
e^{\alpha_{1}\delta\varphi_{1}+\alpha_{2}\delta\pi_{2}+\alpha_{3}\delta\pi_{3}}\rangle$
$\displaystyle B(\beta)=$ $\displaystyle\langle
e^{\beta_{a}\delta\pi_{a}+\beta_{b}\delta\varphi_{b}+\beta_{c}\delta\varphi_{c}}\rangle$
$\displaystyle e^{X}=$ $\displaystyle
e^{\alpha_{1}\hat{\varphi}_{1}+\alpha_{2}\hat{\pi}_{2}+\alpha_{3}\hat{\pi}_{3}}$
$\displaystyle e^{Y}=$ $\displaystyle
e^{\beta_{a}\hat{\pi}_{a}+\beta_{b}\hat{\varphi}_{b}+\beta_{c}\hat{\varphi}_{c}}$
$\displaystyle\delta\varphi_{i}=$
$\displaystyle\hat{\varphi}_{i}-\langle\hat{\varphi}_{i}\rangle\equiv\hat{\varphi}_{i}-\bar{\varphi}_{i}\,,\;\delta\pi_{i}=\hat{\pi}_{i}-\langle\hat{\pi}_{i}\rangle\equiv\hat{\pi}_{i}-\bar{\pi}_{i}\,.$

To clarify, there is no sum over repeated indices, and the objects
$\delta\varphi_{i}$ and $\delta\pi_{i}$ here have nothing to do with the
inhomogeneities in the main text. Note also that our definition of
$A(\alpha),B(\beta),X,Y$ is specific to the brackets we wish to compute. This
is what we meant previously when we claimed that the prescription needs to be
repeated (with a different $A(\alpha)$ and $B(\beta)$) for each distinct
bracket.

By construction, the quantities $A(\alpha)$ and $B(\beta)$ contain
respectively the two Weyl-ordered cubic moments involved in the bracket that
we want to compute

$\displaystyle\alpha_{1}\alpha_{2}\alpha_{3}\langle\delta\varphi_{1}\delta\pi_{2}\delta\pi_{3}\rangle_{W}\subset$
$\displaystyle\langle\sum_{n=0}\frac{1}{n!}(\alpha_{1}\delta\varphi_{1}+\alpha_{2}\delta\varphi_{2}+\alpha_{3}\delta\varphi_{3})^{n}\rangle=A(\alpha)$
$\displaystyle\beta_{a}\beta_{b}\beta_{c}\langle\delta\pi_{a}\delta\pi_{b}\delta\pi_{c}\rangle_{W}\subset$
$\displaystyle\langle\sum_{n=0}\frac{1}{n!}(\beta_{a}\delta\pi_{a}+\beta_{b}\delta\varphi_{b}+\beta_{c}\delta\varphi_{c})^{n}\rangle=B(\beta)\,,$

where the left-hand sides arise from picking out the $n=3$ terms of the sum.
These terms naturally form a symmetrically ordered sum, and the overall
coefficients are exactly $1$ when we write them into a Weyl-ordered notation
$\langle\dots\rangle_{W}$. Thus the
$O(\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c})$ term (with
overall coefficient 1) in $\\{A(\alpha),B(\beta)\\}$ will be exactly the
bracket we want

$\displaystyle\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c}\\{\Delta(\varphi_{1}\pi_{2}\pi_{3}),\Delta(\pi_{a}\varphi_{b}\varphi_{c})\\}\subset_{1}\\{A(\alpha),B(\beta)\\}\,.$

where the subscript of $\subset_{1}$ indicates the overall coefficient is $1$.

Now we wish to expand the exponential products in the bracket
$\\{A(\alpha),B(\beta)\\}$ via the BCH formula. We first split the bracket
between the generating functions $A(\alpha)$ and $B(\beta)$ into four parts

$\displaystyle\\{A(\alpha),B(\beta)\\}=$ $\displaystyle\\{\langle
e^{X}e^{-\bar{X}}\rangle,\langle e^{Y}e^{-\bar{Y}}\rangle\\}$ $\displaystyle=$
$\displaystyle\\{\langle e^{X}\rangle,\langle
e^{Y}\rangle\\}e^{-\bar{X}}e^{-\bar{Y}}$ $\displaystyle+$
$\displaystyle\\{\langle e^{X}\rangle,e^{-\bar{Y}}\\}e^{-\bar{X}}\langle
e^{Y}\rangle$ $\displaystyle+$ $\displaystyle\\{e^{-\bar{X}},\langle
e^{Y}\rangle\\}\langle e^{X}\rangle e^{-\bar{Y}}$ $\displaystyle+$
$\displaystyle\\{e^{-\bar{X}},e^{-\bar{Y}}\\}\langle e^{X}\rangle\langle
e^{Y}\rangle\,,$

where the notation $\bar{X},\bar{Y}$ should be obvious — they are
$\bar{X}=\alpha_{1}\bar{\varphi}_{1}+\alpha_{2}\bar{\pi}_{2}+\dots$ and
analogously for $\bar{Y}$.

We then use the BCH formula to calculate the part that contains averages of
exponentials.

$\displaystyle\\{\langle e^{X}\rangle,\langle e^{Y}\rangle\\}=$
$\displaystyle\frac{1}{i\hbar}\langle e^{X}e^{Y}-e^{Y}e^{X}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{i\hbar}\langle
e^{X+Y+[X,Y]/2}-e^{Y+X+[Y,X]/2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{i\hbar}\langle
e^{X+Y}(e^{i(-i[X,Y]/2)}-e^{-i(-i[X,Y]/2)})\rangle$ $\displaystyle=$
$\displaystyle\langle e^{X+Y}\rangle\frac{2}{\hbar}\sin(-\frac{i}{2}[X,Y])$
$\displaystyle\sin(-\frac{i}{2}[X,Y])=$
$\displaystyle\sin(\pi\hbar(\alpha_{1}\beta_{a}\delta_{1a}-\alpha_{2}\beta_{b}\delta_{2b}-\alpha_{2}\beta_{c}\delta_{2c}-\alpha_{3}\beta_{b}\delta_{3b}-\alpha_{3}\beta_{c}\delta_{3c}))\,.$

The notation of delta functions is $\delta_{1a}=\delta(k_{1}+k_{a})$. Every
line but the last applies to any linear sum of canonical variables as $X,Y$.
The last line is specific to the $\varphi\pi\pi-\pi\varphi\varphi$ bracket.

The two mixed brackets — containing an average of exponentials and
exponentials of averages — can be calculated via (2.1). The first of such
brackets gives

$\displaystyle\\{\langle e^{X}\rangle,e^{-\bar{Y}}\\}=$
$\displaystyle\\{\langle e^{X}\rangle,(-\langle Y\rangle)\\}e^{-\bar{Y}}$
$\displaystyle=$ $\displaystyle\frac{1}{i\hbar}\langle e^{X}[X,-Y]\rangle
e^{-\bar{Y}}$ $\displaystyle=$ $\displaystyle\frac{1}{i\hbar}[X,-Y]\langle
e^{X}\rangle e^{-\bar{Y}}\,.$

In the second and third lines, we used the fact that $Y,X$ are linear in
canonical variables. Similarly, the second mixed bracket gives

$\displaystyle\\{e^{-\bar{X}},\langle e^{Y}\rangle\\}=$ $\displaystyle
e^{-\bar{X}}\\{-\langle X\rangle,\langle e^{Y}\rangle\\}$ $\displaystyle=$
$\displaystyle\frac{1}{i\hbar}e^{-\bar{X}}\langle[-X,Y]e^{Y}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{i\hbar}[-X,Y]\langle e^{Y}\rangle
e^{-\bar{X}}\,.$

Finally, the bracket between two exponentials of averages is most direct

$\displaystyle\\{e^{-\bar{X}},e^{-\bar{Y}}\\}=$ $\displaystyle
e^{-\bar{X}}e^{-\bar{Y}}\\{X,Y\\}=\frac{1}{i\hbar}[X,Y]e^{-\bar{X}}e^{-\bar{Y}}\,.$

Combining these brackets, we arrive at a key result of the BCH-PB prescription

$\displaystyle\\{A(\alpha),B(\beta)\\}=$ $\displaystyle\\{\langle
e^{X-\bar{X}}\rangle,\langle e^{Y-\bar{Y}}\rangle\\}$ $\displaystyle=$
$\displaystyle\langle
e^{X-\bar{X}+Y-\bar{Y}}\rangle\frac{2}{\hbar}\sin(\frac{1}{2i}[X,Y])$
$\displaystyle-\frac{1}{i\hbar}[X,Y]A(\alpha)B(\beta)$ (A.1)
$\displaystyle[X,Y]=$ $\displaystyle 2\pi
i\hbar(\alpha_{1}\beta_{a}\delta_{1a}$
$\displaystyle-\alpha_{2}\beta_{b}\delta_{2b}-\alpha_{2}\beta_{c}\delta_{2c}$
$\displaystyle-\alpha_{3}\beta_{b}\delta_{3b}-\alpha_{3}\beta_{c}\delta_{3c})\,.$

We stress that the formula (A.1) is still general, in the sense that as long
as we adjust how the canonical variables in $X$ and $Y$ are summed, we can use
(A.1) to compute any bracket. The specific form of $[X,Y]$ here is merely a
consequence of our current intention of computing a
$\varphi\varphi\pi-\pi\varphi\varphi$ type bracket. Let us now present some
examples of applying the BCH-PB prescription.

###  A.2 Examples of using the generic BCH-PB formula

Here we use (A.1) to calculate some examples. For simplicity of bookkeeping
orders of $\alpha$ and $\beta$ coefficients, we will write $[X,Y]=2\pi
i\hbar\sum O(\alpha\beta)$ to indicate that it is linear in the coefficient-
product $\alpha\times\beta$.

Example 1: The first example is the original $\varphi\pi^{2}-\pi\varphi^{2}$
bracket that we hoped to compute. This will be a very non-trivial test as the
bracket is one of the most difficult cubic-cubic brackets to compute. It
contains both $\pi$ and $\varphi$ in both slots of the bracket and has a
counterintuitive $O(\hbar^{2})$ constant term.

Here $X,Y$ are the ones specified in the previous section

$\displaystyle A(\alpha)=$ $\displaystyle\langle
e^{\alpha_{1}\delta\varphi_{1}+\alpha_{2}\delta\pi_{2}+\alpha_{3}\delta\pi_{3}}\rangle$
$\displaystyle B(\beta)=$ $\displaystyle\langle
e^{\beta_{a}\delta\pi_{a}+\beta_{b}\delta\varphi_{b}+\beta_{c}\delta\varphi_{c}}\rangle$
$\displaystyle e^{X}=$ $\displaystyle
e^{\alpha_{1}\hat{\varphi}_{1}+\alpha_{2}\hat{\pi}_{2}+\alpha_{3}\hat{\pi}_{3}}$
$\displaystyle e^{Y}=$ $\displaystyle
e^{\beta_{a}\hat{\pi}_{a}+\beta_{b}\hat{\varphi}_{b}+\beta_{c}\hat{\varphi}_{c}}\,.$

As explained before, we already know that the
$O(\alpha_{1}\dots\alpha_{3}\beta_{a}\dots\beta_{3})$ term in the direct
expansion of the generating function in the bracket $\\{A(\alpha),B(\beta)\\}$
is multiplied by the $\varphi\varphi\pi-\pi\varphi\varphi$ bracket that we
want (with an overall coefficient $1$). So now to compute said bracket in
terms of fluctuations $\Delta$, we only need to find all the
$O(\alpha_{1}\dots\alpha_{3}\beta_{a}\dots\beta_{c})$ term on the RHS of
(A.1).

Consider the first line of (A.1)

$\displaystyle\frac{2}{\hbar}\sin(\frac{1}{2i}[X,Y])\langle
e^{X-\bar{X}+Y-\bar{Y}}\rangle=$
$\displaystyle\frac{2}{\hbar}\left[\pi\hbar\sum
O(\alpha\beta)-\frac{1}{3!}(\pi\hbar\sum
O(\alpha\beta))^{3}+\dots\right]\langle e^{X-\bar{X}+Y-\bar{Y}}\rangle$
$\displaystyle=$ $\displaystyle\frac{2}{\hbar}[\pi\hbar\sum
O(\alpha\beta)\frac{1}{4!}\langle(X-\bar{X}+Y-\bar{Y})^{4}\rangle+\dots]$
$\displaystyle-\frac{2}{\hbar}\frac{1}{3!}[\pi^{3}\hbar^{3}(\sum
O(\alpha\beta))^{3}\times 1+\dots]\,.$

We immediately see the constant term arise in the last line. However, not
every term of the last line should be included in the
$\varphi\pi^{2}-\pi\varphi^{2}$ bracket, but only the ones with different
indices, as we are looking for
$O(\alpha_{1}\dots\alpha_{3}\beta_{a}\dots\beta_{c})$ terms. We will get back
to this in a brief moment. Let us fist consider the $1/4!$-line above.

For the $1/4!$-line, not all the terms in the expectation value fit the
indices-requirement — it depends on which terms in $\sum O(\alpha\beta)$
multiplies with the expectation value. For example, for
$\alpha_{1}\beta_{a}\delta_{1a}\subset\sum O(\alpha\beta)$, we would obtain
one of the desired fourth-order fluctuations

$\displaystyle\sum
O(\alpha\beta)\frac{1}{4!}\langle(X-\bar{X}+Y-\bar{Y})^{4}\rangle\supset_{1}$
$\displaystyle\frac{1}{4!}\alpha_{1}\beta_{a}\delta_{1a}\langle[(\alpha_{2}\delta\pi_{2}+\alpha_{3}\delta\pi_{3}$
$\displaystyle+\beta_{b}\delta\varphi_{b}+\beta_{c}\delta\varphi_{c})+(\alpha_{1}\delta\varphi_{1}+\beta_{a}\delta\pi_{a})]^{4}\rangle$
$\displaystyle=$
$\displaystyle\alpha_{1}\beta_{a}\delta_{1a}\frac{1}{4!}\langle(\alpha_{2}\delta\pi_{2}+\alpha_{3}\delta\pi_{3}+\beta_{b}\delta\varphi_{b}+\beta_{c}\delta\varphi_{c})^{4}\rangle+\dots$
$\displaystyle\supset_{1}$
$\displaystyle\alpha_{1}\beta_{a}\delta_{1a}\alpha_{2}\alpha_{3}\beta_{b}\beta_{c}\langle(\delta\pi_{2}\delta\pi_{3}\delta\varphi_{b}\delta\varphi_{c})\rangle_{W}\,,$

where all the “$\supset_{1}$” s imply the relation that the left-hand side
contains the right-hand side with an overall coefficient of $1$. Repeating
this logic, the terms in the $1/4!$-line relevant to our bracket are

$\displaystyle\frac{2}{\hbar}[\pi\hbar\sum
O(\alpha\beta)\frac{1}{4!}\langle(X-\bar{X}+Y-\bar{Y})^{4}\rangle+\dots]$
$\displaystyle\supset
2\pi\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c}\times$
$\displaystyle[\delta_{1a}\Delta(\pi_{2}\pi_{3}\varphi_{a}\varphi_{b})-\delta_{2b}\Delta(\varphi_{1}\pi_{3}\pi_{a}\varphi_{c})-\delta_{2c}\Delta(\varphi_{1}\pi_{3}\pi_{a}\varphi_{b})-(2\leftrightarrow
3)]$

The above contains all the $\Delta^{(4)}$ terms in the
$\varphi\pi^{2}-\pi\varphi^{2}$ bracket with the correct overall coefficients
(modulo the $\alpha_{1}\dots\beta_{c}$ factor) 111111This result matches the
previous one I obtained via the original BCH-PB..

Now consider the $(\sum O(\alpha\beta))^{3}$ line. These terms will help us
find the constant terms. Again, we do not need all the terms in this line. The
sum contains $5$ terms and once cubed we only need the resulting product terms
that contain all distinct indices. The $\alpha_{1}\beta_{a}$ factor must
appear and always come from $\alpha_{1}\beta_{a}\delta_{1a}$, whereas the
$\alpha_{2}\alpha_{3}\beta_{b}\beta_{c}$ can come in different ways. There are
$3\times 2\times 1$ ways of getting
$\alpha_{1}\beta_{a}\delta_{1a}\alpha_{2}\beta_{b}(-\delta_{2b})\alpha_{3}\beta_{c}(-\delta_{3c})$
and
$\alpha_{1}\beta_{a}\alpha_{1a}\alpha_{2}\beta_{c}(-\delta_{2c})\alpha_{3}\beta_{b}(-\delta_{3b})$
respectively. Namely, we need 12 terms in $(\sum O(\alpha\beta)^{3})$.
Specifically, the relevant terms in the $O(\sum O(\alpha\beta)^{3})$ line are

$-\frac{2}{\hbar}\frac{1}{3!}[\pi^{3}\hbar^{3}(\sum O(\alpha\beta))^{3}\times
1+\dots]\supset-2\pi^{3}\hbar^{2}\alpha_{1}\dots\alpha_{3}\beta_{a}\dots\beta_{c}(\delta_{1a}\delta_{2b}\delta_{3c}+\delta_{1a}\delta_{2c}\delta_{3b})\,.$
(A.2)

This concludes the search for relevant terms in the first line of (A.1).

Now consider the second line of (A.1), which gives us the
$\Delta^{(2)}\Delta^{(2)}$ terms. Specifically, the relevant terms are

$\displaystyle-\frac{1}{i\hbar}[X,Y]A(\alpha)B(\beta)$
$\displaystyle\supset_{1}-2\pi\sum
O(\alpha\beta)\frac{1}{2!}\langle(\alpha_{1}\varphi_{1}+\alpha_{2}\pi_{2}+\alpha_{3}\pi_{3})^{2}\rangle\langle(\beta_{a}\pi_{a}+\beta_{b}\varphi_{b}+\beta_{c}\varphi_{c})^{2}\rangle$
$\displaystyle=$
$\displaystyle-2\pi\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c}$
$\displaystyle\times[\delta_{1a}\Delta(\pi_{2}\pi_{3})\Delta(\varphi_{b}\varphi_{c})-(\delta_{2b}\Delta(\varphi_{1}\pi_{3})\Delta(\pi_{a}\varphi_{c})+\delta_{2c}\Delta(\varphi_{1}\pi_{3})\Delta(\pi_{a}\varphi_{b})$
$\displaystyle+(2\leftrightarrow 3))]\,.$

Now we have collected all the possible
$\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c}$ terms of the RHS
of (A.1). Matching this with the LHS — the
$\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{c}\beta_{b}\\{\Delta(\varphi_{1}\pi_{2}\pi_{c}),\Delta(\pi_{a}\varphi_{b}\varphi_{c})\\}$
term — we obtain the full result for the bracket

$\displaystyle\\{\Delta(\varphi_{1}\pi_{2}\pi_{3}),\Delta(\pi_{a}\varphi_{b}\varphi_{c})\\}$
$\displaystyle=$ $\displaystyle
2\pi[\delta_{1a}\Delta(\pi_{2}\pi_{3}\varphi_{a}\varphi_{b})-\delta_{2b}\Delta(\varphi_{1}\pi_{3}\pi_{a}\varphi_{c})-\delta_{2c}\Delta(\varphi_{1}\pi_{3}\pi_{a}\varphi_{b})-(2\leftrightarrow
3)]$
$\displaystyle-2\pi^{3}\hbar^{2}(\delta_{1a}\delta_{2b}\delta_{3c}+\delta_{1a}\delta_{2c}\delta_{3b})$
$\displaystyle-2\pi\times[\delta_{1a}\Delta(\pi_{2}\pi_{3})\Delta(\varphi_{b}\varphi_{c})$
$\displaystyle-\delta_{2b}\Delta(\varphi_{1}\pi_{3})\Delta(\pi_{a}\varphi_{c})-\delta_{2c}\Delta(\varphi_{1}\pi_{3})\Delta(\pi_{a}\varphi_{b})-(2\leftrightarrow
3)]$

One can check that this reduces to the quantum mechanical result derived in
[1] if we set all $2\pi\delta_{ij}$ to one.

Example 2: The second example is the $\varphi^{3}-\pi^{3}$ bracket, but we
will only look for the constant (and counterintuitive) term of the last
equation in (3.11) to show how it arises.

The direct expansion of the exponentials in $\\{A(\alpha),B(\beta)\\}$
contains the desired bracket between the fluctuations

$\\{A(\alpha),B(\beta)\\}\supset_{1}\alpha_{1}\dots\alpha_{3}\beta_{a}\dots\beta_{c}\\{\langle\delta\varphi_{1}\delta\varphi_{2}\delta\varphi_{3}\rangle_{W},\langle\delta\pi_{a}\delta\pi_{b}\delta\pi_{c}\rangle_{W}\\}\,.$
(A.3)

where “$\supset_{1}$” means “contains up to an overall coefficient $1$”.

The constant term of the bracket originates from the $\sin(\dots)$ term in
(A.1). The relevant $O(\alpha_{1}\dots\alpha_{3}\beta_{a}\dots\beta_{c})$
terms are

$\displaystyle\frac{2}{\hbar}\sin[\pi\hbar\sum O(\alpha\beta)]\supset_{1}$
$\displaystyle-\frac{2}{\hbar}(\pi\hbar)^{3}\frac{1}{3!}$
$\displaystyle\times[\alpha_{1}\beta_{a}\delta_{1a}+\alpha_{1}\beta_{b}\delta_{1b}+\alpha_{1}\beta_{c}\delta_{1c}$
$\displaystyle+\alpha_{2}\beta_{a}\delta_{2a}+\alpha_{2}\beta_{b}\delta_{2b}+\alpha_{2}\beta_{c}\delta_{2c}$
$\displaystyle+\alpha_{3}\beta_{a}\delta_{3a}+\alpha_{3}\beta_{b}\delta_{3b}+\alpha_{3}\beta_{c}\delta_{3c}]^{3}$
$\displaystyle\supset_{1}$
$\displaystyle-\frac{2}{\hbar}(\pi\hbar)^{3}\frac{1}{3!}$
$\displaystyle\times[3!\alpha_{1}\beta_{a}\delta_{1a}\alpha_{2}\beta_{b}\delta_{2b}\alpha_{3}\beta_{c}\delta_{3c}$
$\displaystyle+(\text{five more possible pairings between
}1,2,3\leftrightarrow a,b,c)]\,.$

Generalizing this to $3+1$ dimensions, via
$(2\pi)\delta(\dots)\rightarrow(2\pi)^{3}\delta^{3}(\dots)$, we see that this
constant term matches the one in (3.11) when $\hbar=1$.

Example 3. This example shows how “mirror-brackets” — such as
$\\{\Delta(\varphi\varphi\pi),\Delta(\varphi\varphi\pi)\\}$ — are computed.
These brackets vanish trivially in the quantum mechanical case due to the lack
of Fourier-space labels. In the field case, the
$\\{\Delta(\varphi_{1}\varphi_{2}\pi_{3}),\Delta(\varphi_{a}\varphi_{b}\pi_{c})\\}$
bracket is one of the two mirror brackets of the cubic-cubic type. (The other
one is the $\varphi\pi\pi-\varphi\pi\pi$ type bracket.)

The linear sums $X$ and $Y$ in this case are

$\displaystyle X=$
$\displaystyle\alpha_{1}\varphi_{1}+\alpha_{2}\varphi_{2}+\alpha_{3}\pi_{3}\,,\;Y=\beta_{a}\varphi_{b}+\beta_{b}\varphi_{b}+\beta_{c}\pi_{c}\,,\;$
$\displaystyle[X,Y]=$ $\displaystyle 2\pi
i\hbar(\alpha_{1}\beta_{c}\delta_{1c}+\alpha_{2}\beta_{c}\delta_{2c}-\alpha_{3}\beta_{a}\delta_{3a}-\alpha_{3}\beta_{b}\delta_{3b})\,.$

The bracket is contained in

$\\{A(\alpha),B(\beta)\\}\supset_{1}\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c}\\{\langle\delta\varphi_{1}\delta\varphi_{2}\delta\pi_{3}\rangle_{W},\langle\delta\varphi_{a}\delta\varphi_{b}\pi_{c}\rangle_{W}\\}.$
(A.4)

We therefore need to look for
$O(\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c})$ terms.

The $O(\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c})$ terms in
the first line of the BCH-PB formula (A.1) are

$\displaystyle\langle
e^{X-\bar{X}+Y-\bar{Y}}\rangle\frac{2}{\hbar}\sin\left(\frac{1}{2i}[X,Y]\right)=$
$\displaystyle\langle e^{X-\bar{X}+Y-\bar{Y}}\rangle\frac{2}{\hbar}$
$\displaystyle\times\frac{2}{\hbar}[\pi\hbar\sum
O(\alpha\beta)-\frac{1}{3!}(\pi\hbar)^{3}\sum
O(\alpha\beta)^{3}+O(\alpha^{5}\beta^{5})]\,.$

It turns out that the $\sum O(\alpha^{3}\beta^{3})$ terms above excludes
$O(\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c})$ terms because
in the sum the product $\alpha_{1}\alpha_{2}$ always comes with a factor of
$\beta_{c}^{2}$, which is not the desired power of $\beta_{c}$. This is
ultimately a consequence of the specific form of $[X,Y]$ for this bracket. So
we only have

$\displaystyle\langle
e^{X-\bar{X}+Y-\bar{Y}}\rangle\frac{2}{\hbar}\sin\left(\frac{1}{2i}[X,Y]\right)\supset_{1}$
$\displaystyle\langle e^{X-\bar{X}+Y-\bar{Y}}\rangle\frac{2}{\hbar}$
$\displaystyle\times\frac{2}{\hbar}[\pi\hbar\sum O(\alpha\beta)]\,.$

Here the $\sum O(\alpha\beta)\propto[X,Y]$, so we simply need to pick the
remaining $O(\alpha^{2}\beta^{2})$ terms from $\langle
e^{X-\bar{X}+Y-\bar{Y}}\rangle$ (the $n=4$ term to be specific) for
$O(\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c})$. The terms in
the exponential are naturally completely symmetrized and readily give
$\Delta^{(4)}$’s.

In the end, we obtain the relevant terms

$\displaystyle\langle
e^{X-\bar{X}+Y-\bar{Y}}\rangle\frac{2}{\hbar}\sin\left(\frac{1}{2i}[X,Y]\right)\supset_{1}2\pi\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c}$
$\displaystyle[\Delta(\varphi_{2}\pi_{3}\varphi_{a}\varphi_{b})\delta_{1c}+\Delta(\varphi_{1}\pi_{3}\varphi_{a}\varphi_{b})\delta_{2c}$
$\displaystyle-\Delta(\varphi_{1}\varphi_{2}\varphi_{b}\pi_{c})\delta_{3a}-\Delta(\varphi_{1}\varphi_{2}\varphi_{a}\pi_{c})\delta_{3b}]\,.$

The remaining $O(\alpha_{1}\dots\beta_{c})$ terms are in the second line of
(A.1). The factor $[X,Y]$ gives us $O(\alpha\beta)$. We thus need
$O(\alpha^{2})\subset A(\alpha)$ and $O(\beta^{2})\subset A(\beta)$. Picking
out the right terms is straightforward. One eventually finds the remaining
relevant terms to be

$\displaystyle-\frac{1}{i\hbar}[X,Y]A(\alpha)B(\beta)\supset_{1}-\alpha_{1}\alpha_{2}\alpha_{3}\beta_{a}\beta_{b}\beta_{c}(2\pi)$
$\displaystyle[\Delta(\varphi_{2}\pi_{3})\Delta(\varphi_{a}\varphi_{b})\delta_{1c}+\Delta(\varphi_{1}\pi_{3})\Delta(\varphi_{a}\varphi_{b})\delta_{2c}$
$\displaystyle-\Delta(\varphi_{1}\varphi_{2})\Delta(\varphi_{b}\pi_{c})\delta_{3a}-\Delta(\varphi_{1}\varphi_{2})\Delta(\varphi_{a}\varphi_{c})\delta_{3b}]\,.$

Combining the relevant terms in $\langle
e^{X-\bar{X}+Y-\bar{Y}}\rangle\frac{2}{\hbar}\sin\left(\frac{1}{2i}[X,Y]\right)$
and $-\frac{1}{i\hbar}[X,Y]A(\alpha)B(\beta)$, we obtain the results for
$\\{\Delta(\varphi_{1}\varphi_{2}\pi_{3}),\Delta(\varphi_{a}\varphi_{b}\pi_{c})\\}$,
which we list along with the other brackets in the next section.

###  A.3 Poisson brackets for quadratic and cubic fluctuations

Following the derivations presented above, we list all Poisson brackets
between quadratic $\Delta^{(2)}$ and cubic $\Delta^{(3)}$ fluctuations
relevant for the evolution in a $\varphi^{3}$-model. These brackets are
essential for deriving the equations of motion (3.13) and (3.14). It can be
verified that the brackets listed here align with the general bracket matching
rules specified in (3.9), (3.7), and (3.10).

The non-vanishing Poisson brackets between two quadratic fluctuations
$\Delta^{(2)}$ are as follows:

$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}),\Delta(\pi_{\mathbf{a}}\varphi_{\mathbf{b}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{b}})+\tilde{\delta}_{\mathbf{p,a}}\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{b}})$
$\displaystyle\equiv\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{b}})+(\mathbf{k\leftrightarrow
p}),$
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{p}}\pi_{\mathbf{b}})+\tilde{\delta}_{\mathbf{p,a}}\Delta(\varphi_{\mathbf{k}}\pi_{\mathbf{b}})+\tilde{\delta}_{\mathbf{k,b}}\Delta(\varphi_{\mathbf{p}}\pi_{\mathbf{a}})+\tilde{\delta}_{\mathbf{p,b}}\Delta(\varphi_{\mathbf{k}}\pi_{\mathbf{a}})$
$\displaystyle\equiv\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{p}}\pi_{\mathbf{b}})+(\mathbf{k\leftrightarrow
p},\mathbf{a\leftrightarrow b}),$
$\displaystyle\\{\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{p}}),\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})\\}$
$\displaystyle=\Big{[}-\tilde{\delta}_{\mathbf{k,a}}\Delta(\pi_{\mathbf{p}}\pi_{\mathbf{b}})\Big{]}+(\mathbf{k\leftrightarrow
p}),$
$\displaystyle\\{\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{p}}),\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\\}$
$\displaystyle=\Big{[}-\tilde{\delta}_{\mathbf{k,a}}\Delta(\pi_{\mathbf{p}}\varphi_{\mathbf{b}})\Big{]}+(\mathbf{k\leftrightarrow
p})(\mathbf{a\leftrightarrow b}).$

The notation $(\mathbf{k\leftrightarrow p})$ implies a term identical to the
preceding one but with momenta $\mathbf{k}$ and $\mathbf{p}$ interchanged.
Similarly, $(\mathbf{k\leftrightarrow p})(\mathbf{a\leftrightarrow b})$
indicates three additional terms, analogous to the preceding term, with the
pairs of momenta $(\mathbf{k},\mathbf{p})$ interchanged or
$(\mathbf{a},\mathbf{b})$ exchanged, i.e., three new terms by permutating
momenta $(\mathbf{k},\mathbf{p};\mathbf{a},\mathbf{b})$ to
$(\mathbf{p},\mathbf{k};\mathbf{a},\mathbf{b})$,
$(\mathbf{k},\mathbf{p};\mathbf{b},\mathbf{a})$, and
$(\mathbf{p},\mathbf{k};\mathbf{b},\mathbf{a})$.

We now enumerate the non-zero Poisson brackets between a quadratic fluctuation
$\Delta^{(2)}$ and a cubic fluctuation $\Delta^{(3)}$. For brevity, we only
include brackets involving $\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}})$
and $\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{p}})$, as these are the only
fluctuations appearing in the Hamiltonian (3.6) and contributing to the
equations of motion. The relevant Poisson brackets are as follows:

$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}),\Delta(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})+(\mathbf{k\leftrightarrow
b}),$
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{p}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}})+(\mathbf{k\leftrightarrow
p})(\mathbf{a\leftrightarrow b}),$
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{p}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})+(\mathbf{k\leftrightarrow
p})(\mathbf{a\to b\to c}),$
$\displaystyle\\{\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{p}}),\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})\\}$
$\displaystyle=\Big{[}-\tilde{\delta}_{\mathbf{k,a}}\Delta(\pi_{\mathbf{p}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})\Big{]}+(\mathbf{k\leftrightarrow
p})(\mathbf{a\to b\to c}),$
$\displaystyle\\{\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{p}}),\Delta(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})\\}$
$\displaystyle=\Big{[}-\tilde{\delta}_{\mathbf{k,a}}\Delta(\pi_{\mathbf{p}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}})\Big{]}+(\mathbf{k\leftrightarrow
p})(\mathbf{a\leftrightarrow b}),$
$\displaystyle\\{\Delta(\pi_{\mathbf{k}}\pi_{\mathbf{p}}),\Delta(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})\\}$
$\displaystyle=\Big{[}-\tilde{\delta}_{\mathbf{k,a}}\Delta(\pi_{\mathbf{p}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})\Big{]}+(\mathbf{k\leftrightarrow
p}).$

In the above, the notation $(\mathbf{k\to p\to q})$ signifies two additional
terms similar to the preceding term, but with momenta
$(\mathbf{k},\mathbf{p},\mathbf{q})$ cycled to
$(\mathbf{p},\mathbf{q},\mathbf{k})$ and $(\mathbf{q},\mathbf{k},\mathbf{p})$
respectively.

Finally, we write the Poisson brackets between
$\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})$ and
quadratic and cubic fluctuations.

$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(\pi_{\mathbf{a}}\varphi_{\mathbf{b}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}\varphi_{\mathbf{b}})+(\mathbf{k\to
p\to q}),$
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}\pi_{\mathbf{b}})+(\mathbf{k\to
p\to q})(\mathbf{a\leftrightarrow b}),$
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Big{[}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})-\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Delta(\varphi_{\mathbf{b}}\varphi_{\mathbf{c}})\Big{]}+(\mathbf{k\to
p\to q}),$
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Big{[}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}})-\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Delta(\pi_{\mathbf{b}}\varphi_{\mathbf{c}})\Big{]}$
$\displaystyle\quad+(\mathbf{k\to p\to q})(\mathbf{a\leftrightarrow b}),$
$\displaystyle\\{\Delta(\varphi_{\mathbf{k}}\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}),\Delta(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})\\}$
$\displaystyle=\tilde{\delta}_{\mathbf{k,a}}\Big{[}\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})-\Delta(\varphi_{\mathbf{p}}\varphi_{\mathbf{q}})\Delta(\pi_{\mathbf{b}}\pi_{\mathbf{c}})\Big{]}$
$\displaystyle\quad+(\mathbf{k\to p\to q})(\mathbf{a\to b\to c})$
$\displaystyle\quad-\frac{\hbar^{2}}{4}\Big{[}\tilde{\delta}_{\mathbf{k,a}}\tilde{\delta}_{\mathbf{p,b}}\tilde{\delta}_{\mathbf{q,c}}+(\mathbf{k\to
p\to q})(\mathbf{a\leftrightarrow b})\Big{]}.$

The brackets above are enough to compute the evolution of quantum fluctuations
of the $\varphi^{3}$-model in the main text.

##  Appendix B Tree-level dynamics of higher-order fluctuations

In this appendix, we provide an overview of the tree-level dynamics of higher-
order fluctuations, that are $\Delta^{(n)}$ with $n>3$.

We begin by exploring the equations of motion for any $\Delta^{(n)}_{1}$ at
the $\mu$-order. These equations of motion are easy to compute because they
only rely on the $\mu^{0}$-order fluctuations $\Delta^{(n)}_{0}$ of a free
theory and adhere to the following matching rules:

$\displaystyle\dot{\Delta}_{1}(X\varphi_{\mathbf{a}})$
$\displaystyle=\cdots+\Delta_{1}(X\pi_{\mathbf{a}}),$
$\displaystyle\dot{\Delta}_{1}(X\pi_{\mathbf{a}})$
$\displaystyle=\cdots-\omega_{a}^{2}\Delta_{1}(X\varphi_{\mathbf{a}})-\mu\int_{p}\left[\Delta_{0}(X\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})-\Delta_{0}(X)\Delta_{0}(\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})\right],$
$\displaystyle\dot{\Delta}_{1}(X\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})$
$\displaystyle=\cdots+\frac{\mu}{2}\tilde{\delta}_{\mathbf{a,b,c}}\Delta_{0}(X),$

where $\Delta_{0}(\cdots)$ can be computed via Wick’s theorem as in Section
3.4.1.

Under the initial condition $\Delta_{1}(\cdots)=0$, we find that the solutions
to these equations are generally expressed as

$\displaystyle\Delta_{1}(X_{\mathbf{a}_{1}}X_{\mathbf{a}_{2}}X_{\mathbf{a}_{3}}\cdots
X_{\mathbf{a}_{n}})=\sum_{1\leq i<j<k\leq
n}\Delta_{1}(X_{\mathbf{a}_{i}}X_{\mathbf{a}_{j}}X_{\mathbf{a}_{k}})\ \times$
$\displaystyle\Delta_{0}(X_{\mathbf{a}_{1}}\cdots
X_{\mathbf{a}_{i-1}}X_{\mathbf{a}_{i+1}}\cdots
X_{\mathbf{a}_{j-1}}X_{\mathbf{a}_{j+1}}\cdots
X_{\mathbf{a}_{k-1}}X_{\mathbf{a}_{k+1}}\cdots X_{\mathbf{a}_{n}}).$ (B.1)

This indicates that for $\mu$-order $\Delta^{(n)}_{1}$ where $n>3$, there are
no ”non-trivial” terms: All such fluctuations are essentially products of
$\Delta^{(3)}_{1}$ and $\Delta^{(2)}_{0}$.

The evolution of any fluctuation $\Delta^{(n)}_{k}$ involving $n$ fields at
$\mu^{k}$-order can be similarly deduced. For general $\mu^{k}$-order of
$\Delta$,s, the evolution is described by:

$\displaystyle\dot{\Delta}_{k}(X\varphi_{\mathbf{a}})$
$\displaystyle=\cdots+\Delta_{k}(X\pi_{\mathbf{a}}),$
$\displaystyle\dot{\Delta}_{k}(X\pi_{\mathbf{a}})$
$\displaystyle=\cdots-\omega_{a}^{2}\Delta_{k}(X\varphi_{\mathbf{a}})$
$\displaystyle\qquad+\mu\int_{p}\left[\sum_{l=0}^{\left[\frac{k}{2}\right]}\Delta_{k-1-2l}(X)\Delta_{2l}(\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})-\Delta_{k-1}(X\varphi_{\mathbf{p}}\varphi_{\mathbf{a-p}})\right],$
$\displaystyle\dot{\Delta}_{k}(X\pi_{a}\pi_{b}\pi_{c})$
$\displaystyle=\cdots+\frac{\mu}{2}\tilde{\delta}_{\mathbf{a,b,c}}\Delta_{k-1}(X),$

with $\Delta_{k}(\cdots)=0$ for $k>0$ as the initial condition. This implies
the evolution of $\Delta_{k}^{(n)}$ depends on $\Delta_{<k}^{(\leq n+1)}$.
With known solutions for $\mu^{0}$\- and $\mu^{1}$-orders of $\Delta$,s, we
can solve the evolution of $\Delta_{k}^{(n)}$ subsequently precisely.

In Section 3.4.3, we have shown that all $\Delta^{(n)}_{k}$ for $k\geq 2$ are
subject to UV divergences. Recalling the solutions (B.1) for
$\Delta_{1}^{(n)}$, it is conjectured that $\Delta^{(n)}_{k}$ contains many
“trivial” terms, including the products of
$\Delta^{(n^{\prime})}_{k^{\prime}}$ with lower $n^{\prime}<n$ and
$k^{\prime}<k$. These “trivial” terms also contribute to the UV divergences in
the solution of $\Delta_{k}^{(n)}$. The remained “significant” part of
$\Delta_{k}^{(n)}$ should be

$\displaystyle\tilde{\Delta}_{k}(X_{\mathbf{a}_{1}}X_{\mathbf{a}_{2}}X_{\mathbf{a}_{3}}\cdots
X_{\mathbf{a}_{n}})=\Delta_{k}(X_{\mathbf{a}_{1}}X_{\mathbf{a}_{2}}X_{\mathbf{a}_{3}}\cdots
X_{\mathbf{a}_{n}})$
$\displaystyle-\sum_{p=2}^{\left[\frac{n}{2}\right]}\sum_{q=0}^{k}\frac{1}{p!(n-p)!}\sum_{(i_{1}i_{2}\cdots
i_{n})\in
S_{n}}\tilde{\Delta}_{q}(X_{\mathbf{a}_{i_{1}}}X_{\mathbf{a}_{i_{2}}}\cdots
X_{\mathbf{a}_{i_{p}}})\Delta_{k-q}(X_{\mathbf{a}_{i_{p+1}}}X_{\mathbf{a}_{i_{p+2}}}\cdots
X_{\mathbf{a}_{i_{n}}}).$

For example, for $\tilde{\Delta}_{2}^{(4)}$, we have

$\displaystyle\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})$
$\displaystyle=\Delta_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})-\Bigg{\\{}\Big{[}\Delta_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta_{0}(\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})+\Delta_{0}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta_{2}(\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})\Big{]}$
$\displaystyle\hskip 120.0pt+(\mathbf{b\to c\to d})\Bigg{\\}},$
$\displaystyle\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\pi_{\mathbf{d}})$
$\displaystyle=\Delta_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\pi_{\mathbf{d}})-\Big{[}\Delta_{0}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta_{2}(\varphi_{\mathbf{c}}\pi_{\mathbf{d}})+(\mathbf{a\to
b\to c})\Big{]},$
$\displaystyle\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})$
$\displaystyle=\Delta_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})-\Delta_{0}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta_{2}(\pi_{\mathbf{c}}\pi_{\mathbf{d}})-\Delta_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}})\Delta_{0}(\pi_{\mathbf{c}}\pi_{\mathbf{d}}),$
$\displaystyle\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})$
$\displaystyle=\Delta_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})-\Big{[}\Delta_{0}(\pi_{\mathbf{c}}\pi_{\mathbf{d}})\Delta_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}})+(\mathbf{b\to
c\to d})\Big{]},$
$\displaystyle\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})$
$\displaystyle=\Delta_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})-\Bigg{\\{}\Big{[}\Delta_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}})\Delta_{0}(\pi_{\mathbf{c}}\pi_{\mathbf{d}})-\Delta_{0}(\pi_{\mathbf{a}}\pi_{\mathbf{b}})\Delta_{2}(\pi_{\mathbf{c}}\pi_{\mathbf{d}})\Big{]}$
$\displaystyle\hskip 120.0pt+(\mathbf{b\to c\to d})\Bigg{\\}}$

which satisfies the following UV finite equations of motion

$\displaystyle\dot{\tilde{\Delta}}_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})$
$\displaystyle=\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})+\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})+\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}}\varphi_{\mathbf{d}})+\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\pi_{\mathbf{d}}),$
$\displaystyle\dot{\tilde{\Delta}}_{2}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})$
$\displaystyle=-\omega_{a}^{2}\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})+\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})+\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}}\varphi_{\mathbf{d}})$
$\displaystyle+\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\pi_{\mathbf{d}})-\mu\Bigg{[}\frac{1}{\omega_{b}}\Delta_{1}(\varphi_{\mathbf{a+b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})+(\mathbf{b\to
c\to d)}\Bigg{]},$
$\displaystyle\dot{\tilde{\Delta}}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})$
$\displaystyle=-\omega_{a}^{2}\tilde{\Delta}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})-\omega_{b}^{2}\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})+\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\varphi_{\mathbf{d}})$
$\displaystyle+\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}}\pi_{\mathbf{d}})-\mu\Bigg{[}\frac{\Delta_{1}(\varphi_{\mathbf{a+c}}\pi_{\mathbf{b}}\varphi_{\mathbf{d}})+\Delta_{1}(\varphi_{\mathbf{b+c}}\pi_{\mathbf{a}}\varphi_{\mathbf{d}})}{\omega_{c}}+(\mathbf{c\leftrightarrow
d)}\Bigg{]},$
$\displaystyle\dot{\tilde{\Delta}}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\varphi_{\mathbf{d}})$
$\displaystyle=-\omega_{a}^{2}\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\varphi_{\mathbf{d}})-\omega_{b}^{2}\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}}\varphi_{\mathbf{d}})-\omega_{c}^{2}\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}}\varphi_{\mathbf{d}})$
$\displaystyle+\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})-\mu\Bigg{[}\frac{1}{\omega_{d}}\Delta_{1}(\varphi_{\mathbf{a+d}}\pi_{\mathbf{b}}\pi_{\mathbf{c}})+(\mathbf{a\to
b\to c})\Bigg{]},$
$\displaystyle\dot{\tilde{\Delta}}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})$
$\displaystyle=-\omega_{a}^{2}\tilde{\Delta}_{2}(\varphi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})-\omega_{b}^{2}\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\varphi_{\mathbf{b}}\pi_{\mathbf{c}}\pi_{\mathbf{d}})-\omega_{c}^{2}\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\varphi_{\mathbf{c}}\pi_{\mathbf{d}})$
$\displaystyle-\omega_{d}^{2}\tilde{\Delta}_{2}(\pi_{\mathbf{a}}\pi_{\mathbf{b}}\pi_{\mathbf{c}}\varphi_{\mathbf{d}}).$

With the initial condition $\tilde{\Delta}_{2}^{(4)}=0$, the resulting
solutions demonstrate harmonic oscillations. The frequency of these
oscillations is $\omega_{\rm{T}}=\omega_{a}+\omega_{b}+\omega_{c}+\omega_{d}$.

Generally, for $\tilde{\Delta}^{(n)}_{k}$, we conclude that:

  * •

If $0\leq k<n-2$, then $\tilde{\Delta}^{(n)}_{k}=0$, meaning there are no non-
trivial terms in $\tilde{\Delta}^{(n)}_{k}$ for $k<n-2$.

  * •

If $k>n-2$, then $\tilde{\Delta}^{(n)}_{k}$ is UV divergent.

  * •

If $k=n-2$, then $\tilde{\Delta}^{(n)}_{k}$ is finite and bounded. In
particular, for $n>2$, given the vacuum initial condition (3.16),
$\tilde{\Delta}_{n-2}(\varphi_{\mathbf{a}_{1}}\varphi_{\mathbf{a}_{2}}\cdots\varphi_{\mathbf{a}_{n}})$
oscillates harmonically with frequency
$(\omega_{\mathbf{a}_{1}}+\omega_{\mathbf{a}_{2}}+\cdots+\omega_{\mathbf{a}_{n}})$.

The bounded nature of $\Delta^{(n)}_{n-2}$ for $n\geq 2$ is a characteristic
of Minkowski spacetime. In Section 4, we will see that in cosmological de
Sitter spacetime, $\Delta^{3}_{1}$ fluctuations diverge in $\varphi^{3}$
theory as the conformal time $\eta$ approaches $0$.

## References

  * [1] Martin Bojowald and Aureliano Skirzewski  “Effective equations of motion for quantum systems”  In _Reviews in Mathematical Physics_ 18.07 World Scientific, 2006, pp. 713–745
  * [2] Nabila Aghanim et al.  “Planck 2018 results. VI. Cosmological parameters”  In _arXiv preprint arXiv:1807.06209_ , 2018
  * [3] Alan H Guth  “Inflationary universe: A possible solution to the horizon and flatness problems”  In _Physical Review D_ 23.2 APS, 1981, pp. 347
  * [4] Viatcheslav F Mukhanov, Hume A Feldman and Robert Hans Brandenberger  “Theory of cosmological perturbations”  In _Physics reports_ 215.5-6 Elsevier, 1992, pp. 203–333
  * [5] Lev Kofman, Andrei Linde and Alexei A Starobinsky  “Reheating after inflation”  In _Physical Review Letters_ 73.24 APS, 1994, pp. 3195
  * [6] Dmitry Krotov and Alexander M Polyakov  “Infrared sensitivity of unstable vacua”  In _Nuclear Physics B_ 849.2 Elsevier, 2011, pp. 410–432
  * [7] AM Polyakov  “Infrared instability of the de Sitter space”  In _arXiv preprint arXiv:1209.4135_ , 2012
  * [8] Ulf Danielsson  “The quantum swampland”  In _Journal of High Energy Physics_ 2019.4 Springer, 2019, pp. 1–14
  * [9] G Jona-Lasinio  “Relativistic field theories with symmetry-breaking solutions”  In _Nuovo Cimento_ 34.CERN-TH-450 CM-P00056946, 1964, pp. 1790–1795
  * [10] Sidney Coleman and Erick Weinberg  “Radiative corrections as the origin of spontaneous symmetry breaking”  In _Physical Review D_ 7.6 APS, 1973, pp. 1888
  * [11] Roman Jackiw and A Kerman  “Time-dependent variational principle and the effective action”  In _Physics Letters A_ 71.2-3 Elsevier, 1979, pp. 158–162
  * [12] David J Mulryne, David Seery and Daniel Wesley  “Moment transport equations for non-Gaussianity”  In _Journal of Cosmology and Astroparticle Physics_ 2010.01 IOP Publishing, 2010, pp. 024
  * [13] David J Mulryne, David Seery and Daniel Wesley  “Moment transport equations for the primordial curvature perturbation”  In _Journal of Cosmology and Astroparticle Physics_ 2011.04 IOP Publishing, 2011, pp. 030
  * [14] Tanmay Vachaspati and George Zahariade  “Classical-quantum correspondence and backreaction”  In _Physical Review D_ 98.6 APS, 2018, pp. 065002
  * [15] Tanmay Vachaspati and George Zahariade  “Classical-quantum correspondence for fields”  In _Journal of Cosmology and Astroparticle Physics_ 2019.09 IOP Publishing, 2019, pp. 015
  * [16] Martin Bojowald and Ding Ding  “Canonical description of cosmological backreaction”  In _Journal of Cosmology and Astroparticle Physics_ 2021.03 IOP Publishing, 2021, pp. 083
  * [17] Martin Bojowald et al.  “Multi-field inflation from single-field models”  In _Journal of Cosmology and Astroparticle Physics_ 2021.08 IOP Publishing, 2021, pp. 047
  * [18] Martin Bojowald and Suddhasattwa Brahma  “Canonical derivation of effective potentials”  In _arXiv preprint arXiv:1411.3636_ , 2014
  * [19] David Brizuela and Unai Muniain  “A moment approach to compute quantum-gravity effects in the primordial universe”  In _Journal of Cosmology and Astroparticle Physics_ 2019.04 IOP Publishing, 2019, pp. 016
  * [20] David Brizuela and Irene De León  “Mode coupling on a geometrodynamical quantization of an inflationary universe”  In _Journal of Cosmology and Astroparticle Physics_ 2021.07 IOP Publishing, 2021, pp. 054
  * [21] David Brizuela and Tomasz Pawłowski  “Quantum fluctuations and semiclassicality in an inflaton-driven evolution”  In _Journal of Cosmology and Astroparticle Physics_ 2022.10 IOP Publishing, 2022, pp. 080
  * [22] Julien Grain and Vincent Vennin  “Canonical transformations and squeezing formalism in cosmology”  In _Journal of Cosmology and Astroparticle Physics_ 2020.02 IOP Publishing, 2020, pp. 022
  * [23] F Arickx, J Broeckhove, W Coene and P Van Leuven  “Gaussian wave-packet dynamics”  In _International Journal of Quantum Chemistry_ 30.S20 Wiley Online Library, 1986, pp. 471–481
  * [24] Oleg V Prezhdo  “Quantized hamilton dynamics”  In _Theoretical Chemistry Accounts_ 116 Springer, 2006, pp. 206–218
  * [25] Mainak Mukhopadhyay and Tanmay Vachaspati  “Rolling classical scalar field in a linear potential coupled to a quantum field”  In _Physical Review D_ 100.9 APS, 2019, pp. 096018
  * [26] Tanmay Vachaspati and George Zahariade  “Classical-quantum correspondence and Hawking radiation”  In _Journal of Cosmology and Astroparticle Physics_ 2019.04 IOP Publishing, 2019, pp. 013
  * [27] Bekir Baytaş, Martin Bojowald and Sean Crowe  “Effective potentials from semiclassical truncations”  In _Physical Review A_ 99.4 APS, 2019, pp. 042114
  * [28] Bekir Baytaş, Martin Bojowald and Sean Crowe  “Faithful realizations of semiclassical truncations”  In _Annals of Physics_ 420 Elsevier, 2020, pp. 168247
  * [29] Fabrizio Cametti, Giovanni Jona-Lasinio, Carlo Presilla and Fabio Toninelli  “Comparison between quantum and classical dynamics in the effective action formalism”  In _arXiv preprint quant-ph/9910065_ , 1999
  * [30] Xingang Chen  “Primordial non-Gaussianities from inflation models”  In _Advances in Astronomy_ 2010 Hindawi, 2010
  * [31] Jerome Martin and Robert H Brandenberger  “Trans-Planckian problem of inflationary cosmology”  In _Physical Review D_ 63.12 APS, 2001, pp. 123501
  * [32] Steven Weinberg  “Quantum contributions to cosmological correlations”  In _Physical Review D_ 72.4 APS, 2005, pp. 043514
  * [33] Stephen A Fulling  “Nonuniqueness of canonical field quantization in riemannian space-time”  In _Physical Review D_ 7.10 APS, 1973, pp. 2850
  * [34] Juan Maldacena  “Non-Gaussian features of primordial fluctuations in single field inflationary models”  In _Journal of High Energy Physics_ 2003.05 IOP Publishing, 2003, pp. 013
  * [35] Ulf H Danielsson  “Note on inflation and trans-Planckian physics”  In _Physical Review D_ 66.2 APS, 2002, pp. 023511
  * [36] Craig J Copi, Dragan Huterer, Dominik J Schwarz and Glenn D Starkman  “Large-angle anomalies in the CMB”  In _Advances in Astronomy_ 2010 Hindawi, 2010
  * [37] Jaume Garriga and Viatcheslav F Mukhanov  “Perturbations in k-inflation”  In _Physics Letters B_ 458.2-3 Elsevier, 1999, pp. 219–225