&=\E\Biggl[\E\Biggl[ \Biggl( \frac{1}{\vert \mathcal{I}_{1} \vert}\sum_{i \in \mathcal{I}_{1}} \Biggl(\frac{I(d,z)(Y-\mu_d(Z,X))}{ \pi_d (Z,X) \hat \lambda_z(X)^{\mathcal{I}_{2}} \lambda_z(X)}(\lambda_z(X) -  \hat \lambda_z(X)^{\mathcal{I}_{2}})\Biggr) \Biggr)^2 \bigg | I_2 \Biggr]\Biggr] \\
&=\E\Biggl[\Var\Biggl[\frac{1}{|\mathcal{I}_{1}|} \sum_{i \in \mathcal{I}_{1}} \Biggl(\frac{I(d,z)(Y-\mu_d(Z,X))}{ \pi_d (Z,X) \hat \lambda_z(X)^{\mathcal{I}_{2}} \lambda_z(X)}(\lambda_z(X) -  \hat \lambda_z(X)^{\mathcal{I}_{2}})\Biggr) \bigg | \mathcal{I}_{2} \Biggr]\Biggr] \\
&=\frac{1}{|\mathcal{I}_{1}|}\E\Biggl[\Var\Biggl[\frac{I(d,z)(Y-\mu_d(Z,X))}{ \pi_d (Z,X) \hat \lambda_z(X)^{\mathcal{I}_{2}} \lambda_z(X)}(\lambda_z(X) -  \hat \lambda_z(X)^{\mathcal{I}_{2}})\bigg | \mathcal{I}_{2} \Biggr]\Biggr] \\
&=\frac{1}{|\mathcal{I}_{1}|}\E\Biggl[\E\Biggl[ \Biggl(\frac{I(d,z)(Y-\mu_d(Z,X))}{ \pi_d (Z,X) \hat \lambda_z(X)^{\mathcal{I}_{2}} \lambda_z(X)}(\lambda_z(X) -  \hat \lambda_z(X)^{\mathcal{I}_{2}})  \Biggr)^2 \bigg | \mathcal{I}_{2} \Biggr]\Biggr] \\
&\leq \frac{1}{\eta^3 |\mathcal{I}_{1}|} (1-\eta)^2\E\Biggl[\E\Biggl[(Y-\mu_d(Z,X))^2(\lambda_z(X) -  \hat \lambda_z(X)^{\mathcal{I}_{2}})^2 \bigg | \mathcal{I}_{2} \Biggr]\Biggr] \\
&= \frac{1}{\eta^3 |\mathcal{I}_{1}|} (1-\eta)^2\E\Biggl[(Y-\mu_d(Z,X))^2(\lambda_z(X) -  \hat \lambda_z(X)^{\mathcal{I}_{2}})^2 \Biggr] \\
&\leq \frac{1}{\eta^3 |\mathcal{I}_{1}|} (1-\eta)^2\epsilon_d\E\Biggl[(\lambda_z(X) -  \hat \lambda_z(X)^{\mathcal{I}_{2}})^2 \Biggr] = \frac{o_p(1)}{N}
\end{align*}

Again, the second equality follows because the summands are mean zero and independent. The last two inequalities follow from Assumption <ref> and <ref>, the fact that the MSE for the inverse weights decays at the same rate as the MSE for the propensities and the fact that $|\mathcal{I}_{1}| = N/2$. Hence, Part $3$ is $o_p(1/ \sqrt{N})$.

Last, using the $L_1$-norm of Part $4$:

\begin{align*}
&\E\Biggl[\bigg| \frac{1}{\vert \mathcal{I}_{1} \vert}\sum_{i \in \mathcal{I}_{1}} \Biggl( I(d,z)(\hat \mu_d(Z,X)^{\mathcal{I}_{2}}  -\mu_d(Z,X)) \left(\frac{1}{ \pi_d (Z,X) \lambda_z(X)} - \frac{1}{ \hat \pi_d (Z,X)^{\mathcal{I}_{2}} \hat \lambda_z(X)^{\mathcal{I}_{2}}} \right)\Biggr) \bigg| \Biggr] \\
&=  \E\Biggl[\bigg| \frac{1}{\vert \mathcal{I}_{1} \vert}\sum_{i \in \mathcal{I}_{1}} \Biggl( \frac{I(d,z)}{\pi_d(Z,X) \hat \lambda_z(X)^{\mathcal{I}_{2}}}(\hat \mu_d(Z,X)^{\mathcal{I}_{2}}  -\mu_d(Z,X)) \left(\hat \lambda_z(X)^{\mathcal{I}_{2}} - \lambda_z(X) + \hat \pi_d(Z,X)^{\mathcal{I}_{2}} - \pi_d(Z,X)\right)\Biggr) \bigg| \Biggr] \\
&\leq  \E\Biggl[ \frac{1}{\vert \mathcal{I}_{1} \vert}\sum_{i \in \mathcal{I}_{1}} \bigg| \frac{I(d,z)}{\pi_d(Z,X) \hat \lambda_z(X)^{\mathcal{I}_{2}}}\bigg|\bigg|\hat \mu_d(Z,X)^{\mathcal{I}_{2}}  -\mu_d(Z,X)\bigg| \bigg|\hat \lambda_z(X)^{\mathcal{I}_{2}} - \lambda_z(X) + \hat \pi_d(Z,X)^{\mathcal{I}_{2}} - \pi_d(Z,X)\bigg| \Biggr] \\
&= \frac{1}{|\mathcal{I}_{1}|} \sum_{i \in \mathcal{I}_{1}} \E\Biggl[ \bigg| \frac{I(d,z)}{\pi_d(Z,X) \hat \lambda_z(X)^{\mathcal{I}_{2}}}\bigg|\bigg|\hat \mu_d(Z,X)^{\mathcal{I}_{2}}  -\mu_d(Z,X)\bigg| \bigg|\hat \lambda_z(X)^{\mathcal{I}_{2}} - \lambda_z(X) + \hat \pi_d(Z,X)^{\mathcal{I}_{2}} - \pi_d(Z,X)\bigg|  \Biggr] \\
&\leq \frac{1}{|\mathcal{I}_{1}|} \sum_{i \in \mathcal{I}_{1}} \E\Biggl[ \bigg| \frac{I(d,z)}{\pi_d(Z,X) \hat \lambda_z(X)^{\mathcal{I}_{2}}}\bigg|\bigg|\hat \mu_d(Z,X)^{\mathcal{I}_{2}}  -\mu_d(Z,X)\bigg| \left( \bigg|\hat \lambda_z(X)^{\mathcal{I}_{2}} - \lambda_z(X) \bigg| + \bigg|\hat \pi_d(Z,X)^{\mathcal{I}_{2}} - \pi_d(Z,X)\bigg| \right) \Biggr] \\
&=  \E\Biggl[ \bigg| \frac{I(d,z)}{\pi_d(Z,X) \hat \lambda_z(X)^{\mathcal{I}_{2}}}\bigg|\bigg|\hat \mu_d(Z,X)^{\mathcal{I}_{2}}  -\mu_d(Z,X)\bigg| \left(\bigg|\hat \lambda_z(X)^{\mathcal{I}_{2}} - \lambda_z(X) \bigg| + \bigg|\hat \pi_d(Z,X)^{\mathcal{I}_{2}} - \pi_d(Z,X)\bigg| \right) \Biggr] \\
&\leq \frac{(1-\eta)^2}{\eta^2}  \E\Biggl[\bigg|\hat \mu_d(Z,X)^{\mathcal{I}_{2}}  -\mu_d(Z,X)\bigg| \left(\bigg|\hat \lambda_z(X)^{\mathcal{I}_{2}} - \lambda_z(X) \bigg| + \bigg|\hat \pi_d(Z,X)^{\mathcal{I}_{2}} - \pi_d(Z,X)\bigg| \right) \Biggr] \\
&\leq  \E\Biggl[ \left(\hat \mu_d(Z,X)^{\mathcal{I}_{2}}  -\mu_d(Z,X)\right)^2 \Biggr]^{1/2} \left( \E\Biggl[\left(\hat \lambda_z(X)^{\mathcal{I}_{2}} - \lambda_z(X)\right)^2\Biggr]^{1/2} + \E\Biggl[\left(\hat \pi_d(Z,X)^{\mathcal{I}_{2}} - \pi_d(Z,X)\right)^2 \Biggr]^{1/2}\right) \\ &= \frac{o_p(1)}{\sqrt{N}}
\end{align*}

The first inequality follows from Cauchy-Schwarz, the fourth line from the triangle inequality, the sixth line from Assumption <ref> and the last equality from Assumption <ref> and <ref> and the fact that $|\mathcal{I}_{1}| = N/2$. Hence, Part $4$ is $o_p(1/\sqrt{N})$.

Hence, we have shown that:

\begin{align*}
\frac{1}{|\mathcal{I}_{1}|} \sum_{i \in \mathcal{I}_{1}} \hat{\tau}_{d,z}^{\mathcal{I}_{1}} - \frac{1}{|\mathcal{I}_{1}|} \sum_{i \in \mathcal{I}_{1}}  \tau_{d,z}^{\star \mathcal{I}_{1}}(W)= o_p\left(\frac{1}{\sqrt{N}}\right)
\end{align*}

Putting all the parts together, we can conclude that:

\begin{align*}
\sqrt{N}(\hat \tau_{d,z} - \tau_{d,z}) &\\
&= \underbrace{\sqrt{N} \biggl(\frac{\vert \mathcal{I}_{11} \vert}{N} \hat \tau_{d,z}^{\mathcal{I}_{1}} -\frac{\vert \mathcal{I}_{1} \vert}{N} \tau_{d,z}^{\star, \mathcal{I}_{1}} \biggr)}_{=o_p(1)} +
\underbrace{\sqrt{N} \biggl( \frac{\vert \mathcal{I}_{2} \vert}{N} \hat \tau_{d,z}^{\mathcal{I}_{2}} - \frac{\vert \mathcal{I}_{2} \vert}{N} \tau_{d,z}^{\star, \mathcal{I}_{2}} \biggr)}_{=o_p(1)} + \underbrace{\sqrt{N} (\tau_{d,z}^{\star} - \tau_{d,z})}_{\xrightarrow{d} N(0, V^\star)}
\end{align*}

Hence, the estimator $\hat \theta_{l,m,u,v}^C$ is $\sqrt{N}$-consistent and asymptotically normal.

§ APPENDIX: ESTIMATION PROCEDURES

Algorithm <ref> shows how a propensity score can be truncated and normalised for better finite sample properties. This algorithm can be used for all propensity scores in this paper, namely $\lambda_z(W)$, $\lambda_z(X)$, $\pi_d(Z,X)$ and $\omega_{d,z}(X)$. From now on, the propensity score is called $\pi_d(Z,X)$. Be aware that the score function has to be adapted because weight$_{norm, i}$ replaces $\frac{I(d)}{\pi_d(Z,X)}$ and not only $\pi_d(Z,X)$.
Normalisation and truncation of propensity score based weights
Data: $\{(w_i), (z_i), (y_i)\}$

Propensity score: $\pi_d(z_i,x_i)$
Normalised and truncated weights:  weight$_{norm,i}$

Number of observations: $N$

i = 1, …, N
$\pi_d(z_i,x_i) = \max(\pi_d(z_i,x_i),0.0001)$

Definition weights:

$weight_i = \frac{I(d)}{\pi_d(z_i,x_i)}$

Define: Sum of the weights: $weight_{sum} = \sum_{i = 1}^N weight_i$

i = 1, …, N

$weight_{norm,i} = \frac{weight_i}{weight_{sum}}$

$weight_{norm,i} = \min(weight_{norm,i}, 0.05)$

Define: Sum of the new weights: $weight_{norm,sum} = \sum_{i = 1}^N weight_{norm,i}$

i = 1, …, N

$weight_{norm,i} = \frac{weight_{norm,i}}{weight_{norm,sum} } \cdot N$

Algorithm <ref> shows the algorithm to estimate the $\Delta$BGATE with discrete treatment and moderator variables.

DML for $\Delta$BGATE
Data: $\{(x_i), (z_i), (d_i), (y_i)\}$
$\Delta$BGATE = $\hat{\theta}_{l,m,u,v}^{\Delta B}$

create folds: Split sample into K random folds $(S_k)_{k=1}^K$ of observations $\{1, \dots, N\}$ with size of each fold $N/K$. Define $S_k^c \coloneqq \{1, \dots, N\} \backslash \{S_k\}$

k in $\{1, \dots,K\}$
d in $\{l, m\}$
Response Functions:

estimate: $\mu_d(z_i,x_i) = \E[Y |D = d, X = x_i, Z = z_i]$ in $\{(x_i) (y_i)\}_{i \in S_k^c, d_i = d}$

Propensity Score:

estimate: $\pi_d(x_i,z_i) = P(D = d | X = x_i, Z = z_i)$ in $\{(x_i), (d_i), (z_i)\}_{i \in S_k^c}$

estimate: $\hat{\delta}_{l,m}(z_i,x_i) = \hat{\mu}_l(z_i,x_i) - \hat{\mu}_m(z_i, x_i) + \frac{I(l)(y_i - \hat{\mu}_l(z_i, x_i))}{\hat{\pi}_l(x_i,z_i)} - \frac{I(m)(y_i - \hat{\mu}_m(z_i,x_i))}{\hat{\pi}_m(x_i,z_i)}$ in $\{(y_i), (x_i), (d_i), (z_i)\}_{i \in S_k}$

create folds: Split sample $S_k$ into J random folds $(S_j)_{j=1}^J$ of observations $\{1, \dots, J\}$ with size of each fold $N/(K \cdot J)$. Define $S_{j}^c \coloneqq \{1, \dots, J\} \backslash \{S_j\}$

j in $\{1, \dots,J\}$
z in $\{v, u\}$
Pseudo-outcome regression:

estimate: $g_z(w_i) = \E[\hat{\delta}_{l,m}(Z,X) | Z =z, W = w_i]$ in $\{(w_i)\}_{i \in S_j^c, z_i = z}$

Propensity score:

estimate: $\lambda_z(w_i) = P(Z =z | W = w_i)$ in  $\{(w_i), (z_i)\}_{i \in S_j^c}$
$\Delta$BGATE Function:

estimate: $\hat{\theta}_{j,l,m,u,v}^{\Delta B}= \frac{1}{N/(K \cdot J)}\sum_{i \in S_j} \Bigl[\hat{g}_u(w_i) - \hat{g}_v(w_i) + \frac{I(u)(\hat{\delta}_{l,m}(z_i,x_i) - \hat{g}_u(w_i))}{\hat{\lambda}_u(w_i)} -\frac{I(v)(\hat{\delta}_{l,m}(z_i,x_i) - \hat{g}_v(w_i))}{\hat{\lambda}_v(w_i)} \Bigr]$

Standard errors:

estimate: $\hat{\theta}_{l,m,u,v,j}^{\Delta B_{SE}}= \frac{1}{N/(K \cdot J)}\sum_{i \in S_j} \Bigl[\Bigl(\hat{g}_u(w_i) - \hat{g}_v(w_i) + \frac{I(u)(\delta_{l,m}(z_i,w_i) - \hat{g}_u(w_i))}{\hat{\lambda}_u(w_i)} -\frac{I(v)(\hat{\delta}_{l,m}(z_i,x_i) - \hat{g}_v(w_i))}{\hat{\lambda}_v(w_i)} \Bigr)^2\Bigr]$

estimate effect: $\hat{\theta}_{l,m,u,v}^{\Delta B} = \frac{1}{J\cdot K}\sum_{j = 1}^J \sum_{k = 1}^K \hat{\theta}^{\Delta B}_{j,k,l,m,u,v}$

estimate standard errors: $SE(\hat{\theta}_{l,m,u,v}^{\Delta B}) = \sqrt{\frac{1}{J \cdot K}\sum_{j = 1}^J \sum_{k = 1}^K \hat{\theta}_{j,k,l,m,u,v}^{\Delta B_{SE}} - \hat{\theta}_{j,k,l,m,u,v}^{\Delta B}\hat{\theta}_{j,k,l,m,u,v}^{\Delta B}}$

Algorithm <ref> shows the suggested estimation procedure for the $\Delta$CBGATE. It is similar to the estimation of an ATE but with a different score function.
DML for $\Delta$CBGATE
Data: $\{(x_i), (z_i), (d_i), (y_i)\}$
$\Delta$CBGATE = $\hat{\theta}^{\Delta C}$

create folds: Split sample into K random folds $(S_k)_{k=1}^K$ of observations $\{1, \dots, N\}$ with size of each fold $N/K$. Also define $S_k^c \coloneqq \{1, \dots, N\} \backslash S_k$

k in $\{1, \dots,K\}$
d in $\{l, m\}$
z in $\{v, u\}$
Response Functions:

estimate: $\mu_{d,z}(x_i) = \E[Y | X = x_i, D = d, Z = z]$ in $\{(x_i) (y_i)\}_{i \in S_k^c, d_i = d, z_i = z}$

Propensity Score:

Version 1:

estimate: $\omega_{d,z}(x_i) = P(D = d, Z = z | X = x_i)$ in $\{(x_i), (d_i), (z_i) \}_{i \in S_k^c}$

Version 2:

estimate: $\pi_d(x_i,z_i) = P(D = d | X = x_i, Z = z_i)$ in $\{(x_i), (d_i), (z_i)\}_{i \in S_k^c}$

estimate: $\lambda_z(x_i) = P(Z = z | X = x_i)$ in $\{(x_i), (d_i), (z_i) \}_{i \in S_k^c}$

calculate: $\omega_{d,z}(x_i) = \pi_d(x_i,z_i) \cdot \lambda_z(x_i)$

$\Delta$CBGATE Function:

estimate: $\hat{\theta}_k^{\Delta C}= \frac{1}{N/K}\sum_{i \in S_k} \Bigl[\hat{\mu}_{l,v}(x_i) - \hat{\mu}_{m,b}(x_i) - \hat{\mu}_{l,u}(x_i) + \hat{\mu}_{m,u}(x_i) + \frac{I(l,v)(y_i - \hat{\mu}_{l,v}(x_i))}{\hat{\omega}_{l,v}(x_i)} - \frac{I(m,v)(y_i - \hat{\mu}_{m,v}(x_i))}{\hat{\omega}_{m,v}(x_i)} -\frac{I(l,u)(y_i - \hat{\mu}_{l,u}(x_i))}{\hat{\omega}_{l,u}(x_i)} + \frac{I(m,u)(y_i - \hat{\mu}_{m,u}(x_i))}{\hat{\omega}_{m,u}(x_i)}\Bigr]$

Standard errors:

estimate: $\hat{\theta}_k^{\Delta C, SE}= \frac{1}{N/K}\sum_{i \in S_k} \Bigl[\Bigl(\hat{\mu}_{l,v}(x_i) - \hat{\mu}_{m,v}(x_i) - \hat{\mu}_{l,u}(x_i) + \hat{\mu}_{m,u}(x_i) + \frac{I(l,v)(y_i - \hat{\mu}_{l,v}(x_i))}{\hat{\omega}_{l,v}(x_i)} - \frac{I(m,v)(y_i - \hat{\mu}_{m,v}(x_i))}{\hat{\omega}_{m,v}(x_i)} -\frac{I(l,u)(y_i - \hat{\mu}_{l,u}(x_i))}{\hat{\omega}_{l,u}(x_i)} + \frac{I(m,u)(y_i - \hat{\mu}_{m,u}(x_i))}{\hat{\omega}_{m,u}(x_i)}\Bigr)^2\Bigr]$

estimate effect: $\hat{\theta}^{\Delta C} = \frac{1}{K}\sum_{k = 1}^K \hat{\theta}^{\Delta C}_k$

estimate standard errors: $SE(\hat{\theta}^{\Delta C}) = \sqrt{\frac{1}{K}\sum_{k = 1}^K \hat{\theta}_k^{\Delta C, SE} - \hat{\theta}^{\Delta C}\hat{\theta}^{\Delta C}}$

Algorithm <ref> shows the procedure for estimating $\tau$ with DML by . Summarised, estimate an average treatment effect (ATE) in groups $Z = 1$ and $Z = 0$ separately. Then take the difference of those two effects to receive $\hat{\tau}$.

DML for $\Delta$GATE
Data: $\{(x_i), (d_i), (z_i), (y_i)\}$
$GATE_u$ - $GATE_v$: $\hat{\theta}^{\Delta G}$

z in $\{v, u\}$
create folds: Split sample into K random folds $(S_k)_{k=1}^K$ of observations $\{1, \dots, N\}$ with size of each fold $N/K$. Also define $S_k^c \coloneqq \{1, \dots, N\} \backslash S_k$

k in $\{1, \dots,K\}$
d in $\{l, m\}$
Response Functions:

estimate: $\mu_d(x_i, z_i) = \E[Y | D = d, X = x_i, Z = z_i]$ in $\{(x_i), (z_i), (y_i)\}_{i \in S_k^c, d_i = d}$

Propensity Score:

estimate: $\pi_d(x_i,z_i) = P(D = d | X = x_i, Z = z_i)$ in $\{(x_i), (d_i), (z_i) \}_{i \in S_k^c}$

ATE Function:

estimate: $\hat{\tau}_k=\frac{1}{N/K}\sum_{i \in S_k} \Bigl[\hat{\mu}_l(x_i, z_i) - \hat{\mu}_m(x_i, z_i) + \frac{\mathcal{I}(d_i = l)(y_i - \hat{\mu}_l(x_i, z_i))}{\hat{\pi}_l(x_i, z_i)} - \frac{\mathcal{I}(d_i = m)(y_i - \hat{\mu}_m(x_i, z_i))}{\hat{\pi}_m(x_i, z_i)}\Bigr]$

Standard errors:

estimate: $\hat{\tau}_k^{SE}=\frac{1}{N/K}\sum_{i \in S_k} \Bigl[\Bigl(\hat{\mu}_l(x_i, z_i) - \hat{\mu}_m(x_i, z_i) + \frac{\mathcal{I}(d_i = l)(y_i - \hat{\mu}_l(x_i, z_i))}{\hat{\pi}_l(x_i, z_i)} - \frac{\mathcal{I}(d_i = m)(y_i - \hat{\mu}_m(x_i, z_i))}{(\hat{\pi}_m(x_i, z_i))}\Bigr)^2\Bigr]$
estimate effect: $\hat{\tau}_z = \frac{1}{K}\sum_{k = 1}^K \hat{\tau}_k$

estimate standard errors: $\Var(\hat{\tau}_z) = \frac{1}{K}\sum_{k = 1}^K \hat{\tau}_k^{SE} - \hat \tau_z^2$

calculate effect: $\hat{\theta}_{l,m,u,v}^{\Delta G} = \hat{\tau}_v - \hat{\tau}_u$

calculate standard errors: $SE(\hat{\theta}_{l,m,u,v}^{\Delta G}) = \sqrt{\Var(\hat{\tau}_u) + \Var(\hat{\tau}_v)}$

§ APPENDIX: MONTE CARLO STUDY

This section explains details of the Monte Carlo study conducted for the $\Delta$BGATE and the $\Delta$CBGATE. For simplicity, the simulations cover the case where $Z$ and $D$ are binary variables.

§.§ Performance measures

For the evaluation of the different estimators, different measures are considered. First, we look at the measures to evaluate the estimation of the effects. The performance measures are shown for $\hat{\theta}_{l,m,u,v}^{\Delta B}$ only, but the same performance measures are also used for $\hat{\theta}_{l,m,u,v}^{\Delta C}$ and $\hat{\theta}_{l,m,u,v}^{\Delta G}$.

\begin{align*}
BIAS(\hat{\theta}_{l,m,u,v}^{\Delta B}) &= \frac{1}{R}\sum_{r = 1}^R\hat{\theta}_{r,l,m,u,v}^{\Delta B} - \theta_{l,m,u,v}^{\Delta B} \\
|BIAS(\hat{\theta}_{l,m,u,v}^{\Delta B})| &= \frac{1}{R}\sum_{r = 1}^R | \hat{\theta}_{r,l,m,u,v}^{\Delta B} - \theta_{l,m,u,v}^{\Delta B}| \\
SD(\hat{\theta}_{l,m,u,v}^{\Delta B}) &= \sqrt{\frac{1}{R}\sum_{r = 1}^R(\hat{\theta}_{r,l,m,u,v}^{\Delta B} - \frac{1}{R}\sum_{r = 1}^R \hat{\theta}_{r,l,m,u,v}^{\Delta B})^2} \\
RMSE(\hat{\theta}_{l,m,u,v}^{\Delta B}) &= \sqrt{\frac{1}{R} \sum_{r = 1}^R(\hat{\theta}_{r,l,m,u,v}^{\Delta B}- \theta_{l,m,u,v}^{\Delta B})^2} \\
SKEWNESS(\hat{\theta}_{l,m,u,v}^{\Delta B}) &= \frac{1}{R} \sum_{i = 1}^R \left( \frac{\hat{\theta}_{l,m,u,v}^{\Delta B} - \frac{1}{R}\sum_{r = 1}^R \hat{\theta}_{r,l,m,u,v}^{\Delta B}}{SD(\hat{\theta}_{l,m,u,v}^{\Delta B})}\right)^3 \\
Ex.KURTOSIS(\hat{\theta}_{l,m,u,v}^{\Delta B}) &= \frac{1}{R} \sum_{i = 1}^R \left( \frac{\hat{\theta}_{r,l,m,u,v}^{\Delta B} - \frac{1}{R}\sum_{r = 1}^R \hat{\theta}_{r,l,m,u,v}^{\Delta B}}{SD(\hat{\theta}_{l,m,u,v}^{\Delta B})}\right)^4 - 3
\end{align*}

Furthermore, the following performance measures are used to evaluate the inference procedures:

\begin{align*}
BIAS(\widehat{SE}(\hat{\theta}_{l,m,u,v}^{\Delta B})) = \frac{1}{R} &\sum_{i = 1}^R SE^r(\hat{\theta}_{r,l,m,u,v}^{\Delta B}) - SD(\hat{\theta}_{l,m,u,v}^{\Delta B}) \\
CovP(\alpha) = P\Biggl[&\left(\theta_{l,m,u,v}^{\Delta B} > \frac{1}{R}\sum_{i = 1}^r(\hat{\theta}_{r,l,m,u,v}^{\Delta B} - Z_{\frac{\alpha}{2}}SE(\hat{\theta}_{r,l,m,u,v}^{\Delta B})) \right) \\ \cup &\left(\theta_{l,m,u,v} < \frac{1}{R}\sum_{i = 1}^r(\hat{\theta}_{r,l,m,u,v}^{\Delta B} + Z_{1 -\frac{\alpha}{2}}SE(\hat{\theta}_{r,l,m,u,v}^{\Delta B}))\right)\Biggr]
\end{align*}

§.§ Details for $\Delta$BGATE

§.§.§ Random Forest tuning parameters

The optimal hyperparameters for the different random forests, namely the tree depth and the minimum leaf size, are tuned by a grid search (maximum tree depth: [5, 10, 15, 20], minimum leaf size: [2, 5, 10, 20]) using a random forest with 1000 trees and three folds. They are tuned using five different data draws and then fixed for the whole simulation. The optimal combination of maximal tree depth and minimum leaf size is chosen by taking the combination that appears most often in the five draws. Table <ref> shows the optimal hyperparameters for the different DGPs and sample sizes.

Simulation Study: Optimal hyperparameters for random forests

max width=

13cDGP: non-linear with linear heterogeneous treatment effects
6cN = 2,500    6cN = 10,000
$\mu_1$     $\mu_0$     $\pi_d$     $g_1$     $g_0$     $\lambda_z$     $\mu_1$     $\mu_0$     $\pi_d$     $g_1$     $g_0$     $\lambda_z$

Number of trees     1,000     1,000     1,000     1,000    1,000    1,000     1,000     1,000     1,000     1,000    1,000    1,000

Maximal tree depth     5     5     5     10     5     5     10     5     5     5     5     5

Minimum leaf size     10     2     2     20     20     20     20     2     20     20     5     20
13cDGP: non-linear with non-linear heterogeneous treatment effects
6cN = 2,500    6cN = 10,000
$\mu_1$     $\mu_0$     $\pi_d$     $g_1$     $g_0$     $\lambda_z$     $\mu_1$     $\mu_0$     $\pi_d$     $g_1$     $g_0$     $\lambda_z$

Number of trees     1,000     1,000     1,000     1,000    1,000    1,000     1,000     1,000     1,000     1,000    1,000    1,000

Maximal tree depth      10     5     5     5     10     5     10     5     5     5     5     5

Minimum leaf size     5     2     2     20     20     20      10     2     20      20     2     20
13cDGP: non-linear with non-linear heterogeneous treatment effects and $Z$ influencing $X_5$
6cN = 2,500    6cN = 10,000
$\mu_1$     $\mu_0$     $\pi_d$     $g_1$     $g_0$     $\lambda_z$     $\mu_1$     $\mu_0$     $\pi_d$     $g_1$     $g_0$     $\lambda_z$

Number of trees     1,000     1,000     1,000     1,000    1,000    1,000     1,000     1,000     1,000     1,000    1,000    1,000

Maximal tree depth     10     5     5     5     10     5     10     5     5     5     5     5

Minimum leaf size     10     2     20     20     20     20     10     2     20     20     20     20

* Note: This table depicts the optimal hyperparameters for the random forests. The grid values are as follows: maximum tree depth: [5, 10, 15, 20], minimum leaf size: [2, 5, 10, 20].

§.§.§ Results

Table <ref> and Table <ref> show the results for a non-linear DGP with linear heterogeneity with $N = 2,500$ and $N = 10,000$, respectively. The first part of the tables depicts the results of estimating the outcome regression in the second estimation step with a linear regression. The second part of the tables shows the results of using a random forest to estimate the outcome regressions in the second step. Using a linear regression works well when the heterogeneities are linear. However, using a random forest works equally well. Furthermore, the tables show that the standard error halves by increasing the sample size by four. This suggests a $\sqrt{N}$-convergence rate of the estimator also in finite samples.

Simulation results: Non-linear DGP with a linear treatment effect (N $= 2,500$)

max width=

11cDGP: non-linear with linear heterogeneous treatment effects
7lEstimation of effects    3lEstimation of standard errors
2nd step     Truth     Bias     |Bias|     Std     RMSE     Skew.     Ex. Kurt.    Bias (SE)     CovP (95)     CovP (80)

(in %)     (in %)
11c2nd estimation step with linear regression
BGATE_linear_linear_2500_1000.tex    11c2nd estimation step with random forest

* Note: This table shows results for $N = 2,500$, $R = 1,000$ and a non-linear DGP with linear heterogeneous treatment effects in two blocks: (1) $\Delta$BGATE results with linear regression in the second step (2) $\Delta$BGATE results with random forest in the second step. $\theta^{\Delta G}$ shows the difference between two GATEs. Column (1) shows the effect estimated, and column (2) shows the second estimation step. The remaining ten columns depict performance measures explained in <ref>.

Simulation results: Non-linear DGP with a linear treatment effect (N $= 10,000$)

max width=

11cDGP: non-linear with linear heterogeneous treatment effects
7lEstimation of effects    3lEstimation of standard errors
2nd step     Truth     Bias     |Bias|     Std     RMSE     Skew.     Ex. Kurt.    Bias (SE)     CovP (95)     CovP (80)

(in %)     (in %)
11c2nd estimation step with linear regression
BGATE_linear_linear.tex    11c2nd estimation step with random forest

* Note: This table shows results for $N = 10,000$, $R = 250$ and a non-linear DGP with linear heterogeneous treatment effects in two blocks: (1) $\Delta$BGATE results with linear regression in the second step (2) $\Delta$BGATE results with random forest in the second step. $\theta^{\Delta G}$ shows the difference between two GATEs. Column (1) shows the effect estimated, and column (2) shows the second estimation step. The remaining ten columns depict performance measures explained in <ref>.

Table <ref> and Table <ref> show the result for a non-linear DGP with non-linear heterogeneous treatment effects. Using a linear regression to estimate the outcome regressions in the second step leads to biased results if we want to balance the distribution of the $X$'s that lead to those heterogeneous treatment effects. This is the case because the heterogeneities are non-linear. However, due to the double-robust property of the DML estimator, the bias is not huge. Using DML with a random forest in the second step works well. Again, the standard errors are halved by increasing the sample size by four.

Simulation results: Non-linear DGP with a non-linear treatment effect (N $= 2,500$)

max width=

11cDGP: non-linear with non-linear heterogeneous treatment effects
7lEstimation of effects    3lEstimation of standard errors
2nd step     Truth     Bias     |Bias|     Std     RMSE     Skew.     Ex. Kurt.    Bias (SE)     CovP (95)     CovP (80)

(in %)     (in %)
11c2nd estimation step with linear regression
BGATE_nonlinear_linear_2500_2000.tex    11c2nd estimation step with random forest

* Note: This table shows results for $N = 2,500$, $R = 1,000$ and a non-linear DGP with a non-linear treatment effect in two blocks: (1) $\Delta$BGATE results with linear regression in the second step (2) $\Delta$BGATE results with random forest in the second step. $\theta^{\Delta G}$ shows the difference between two GATEs. Column (1) shows the effect estimated, and column (2) shows the second estimation step. The remaining ten columns depict performance measures explained in <ref>.

Simulation results: Non-linear DGP with a non-linear treatment effect (N $= 10,000$)

max width=

11cDGP: non-linear with non-linear heterogeneous treatment effects
7lEstimation of effects    3lEstimation of standard errors
2nd step     Truth     Bias     |Bias|     Std     RMSE     Skew.     Ex. Kurt.    Bias (SE)     CovP (95)     CovP (80)

(in %)     (in %)
11c2nd estimation step with linear regression
BGATE_nonlinear_linear.tex    11c2nd estimation step with random forest

* Note: This table shows results for $N = 10,000$, $R = 250$ and a non-linear DGP with a non-linear treatment effect in two blocks: (1) $\Delta$BGATE results with linear regression in the second step (2) $\Delta$BGATE results with random forest in the second step. $\theta^{\Delta G}$ shows the difference between two GATEs. Column (1) shows the effect estimated, and column (2) shows the second estimation step. The remaining ten columns depict performance measures explained in <ref>.

Table <ref> and Table <ref> show the result for a non-linear DGP with non-linear heterogeneous treatment effects and some covariates that are influenced by the moderator variable $Z$. Similarly, as seen in the setting with only non-linear heterogeneous treatment effects, using a linear regression in the second step leads to biased results. Using a random forest in the second step works well, even if the moderator variable influences some confounders.

Simulation results: Non-linear DGP with a non-linear treatment effect and a moderator influencing some covariates (N $= 2,500$)

max width=

11cDGP: non-linear with non-linear treatment effect and Z influencing $X_5$
7lEstimation of effects    3lEstimation of standard errors
2nd step     Truth     Bias     |Bias|     Std     RMSE     Skew.     Ex. Kurt.    Bias (SE)     CovP (95)     CovP (80)

(in %)     (in %)
11c2nd estimation step with linear regression
BGATE_nonlinear_z_linear_2500_2000.tex    11c2nd estimation step with random forest

* Note: This table shows results for $N = 2,500$, $R = 1,000$ for a non-linear DGP with non-linear heterogeneous treatment effects and a covariate influenced by the moderator variable in two blocks: (1) $\Delta$BGATE results with linear regression in the second step (2) $\Delta$BGATE results with random forest in the second step.  $\theta^{\Delta G}$ shows the difference between two GATEs.  Column (1) shows the effect estimated, and column (2) shows the second estimation step. The remaining ten columns depict performance measures explained in <ref>.

Simulation results: Non-linear DGP with a non-linear treatment effect and a moderator influencing some covariates (N $= 10,000$)

max width=

11cDGP: non-linear with non-linear treatment effect and Z influencing $X_5$
7lEstimation of effects    3lEstimation of standard errors
2nd step     Truth     Bias     |Bias|     Std     RMSE     Skew.     Ex. Kurt.    Bias (SE)     CovP (95)     CovP (80)

(in %)     (in %)
11c2nd estimation step with linear regression
BGATE_nonlinear_z_linear.tex    11c2nd estimation step with random forest

* Note: This table shows results for $N = 10,000$, $R = 250$ for a non-linear DGP with non-linear heterogeneous treatment effects and a covariate influenced by the moderator variable in two blocks: (1) $\Delta$BGATE results with linear regression in the second step (2) $\Delta$BGATE results with random forest in the second step.  $\theta^{\Delta G}$ shows the difference between two GATEs.  Column (1) shows the effect estimated, and column (2) shows the second estimation step. The remaining ten columns depict performance measures explained in <ref>.

§.§ Details for $\Delta$CBGATE

§.§.§ Data generating process (DGP)

The DGP is nearly identical to the one used in the first simulation study for the $\Delta$BGATE. We start with simulating a p-dimensional covariate matrix $X_p$ with p=6. The first two covariates are drawn from a uniform distribution $X_{1}, X_{2} \sim \mathcal{U}[0,1]$ and the remaining covariates from a normal distribution $X_{3} \dots, X_{p} \sim \mathcal{N}(0.5, \sqrt{1/12})$. All covariates have a mean of 0.5 and a standard deviation of $\sqrt{1/12}$. In the simulation design that features a correlation between the moderator $Z$ and some of the covariates $X$, the moderator variable is created like in the simulation study for the $\Delta$BGATE. Hence the moderator variable $Z$ is drawn from a Bernoulli distribution with probability $p(Z|X_0, X_1) =  (\alpha  + \gamma \cdot \beta(f(x);2,4))$[$\beta(f(x);2,4)$ denotes a draw from the CDF of a beta distribution with the shape parameters $\alpha = 2$ and $\beta$ = 4.] with $f(x) = X_0 * X_1$, $\alpha=0.1$ and $\gamma=0.8$. The propensity score is created similarly as in <cit.> and <cit.>. In the second simulation design, there is no correlation between the moderator $Z$ and some $X$. We draw noise $\epsilon \sim  \mathcal{N}(0,1)$. If $\epsilon > 0.5$, the moderator variable takes the value 1, and otherwise 0. The treatment variable $D$ is drawn from a Bernoulli distribution with probability $p(D|X_0, X_1, X_5) = (\alpha  + \gamma \cdot\beta(f(x,z); 2, 4))$ with $f(x,z) = \frac{X_0 + X_1 + X_2 + X_5 + Z}{5}$, $\alpha=0.2$ and $\gamma = 0.6$.

Next, the response functions under treatment and non-treatment and the two states of the moderator variable are specified. The non-treatment response function is specified similarly as in <cit.> and creates a difficult non-linear setting. They are given by

\begin{align*}
\mu_{0,1}(X) = \sin(\pi \times X_0 \times X_1) + (X_2 - 0.5)^2 + 0.1X_3 + 0.3X_5 \\
\mu_{0,0}(X) = \sin(\pi \times X_0 \times X_1) + (X_2 - 0.5)^2 + 0.1X_3 + 0.3X_5
\end{align*}

The response functions under treatment are defined differently, namely as:

\begin{align*}
\mu_{1,1}(X) = \mu_{0}(1,X) + \sin(4.9X_0) + \sin(2X_1) + 0.7X_2^4 + 0.4X_5 \\
\mu_{1,0}(X) =\mu_{0}(0,X) + \sin(1.4X_0) + \sin(6X_1) + 0.6X_2^2 + 0.3X_5.
\end{align*}

In contrast to the response functions in the $\Delta$BGATE simulation, they are not directly influenced by the moderator $Z$ because it would violate Assumption <ref>. Hence, it would not be possible to identify a $\Delta$CBGATE. Moreover, we restrict this simulation to non-linear heterogeneitey.

Last, we simulate the potential outcomes as $Y_{d,z} = \mu_{d,z}(X)  + \epsilon_{d,z}$ with noise $\epsilon_{d,z}  \sim \mathcal{N}(0,1)$. Summing up, the data consists of an observable quadruple $(y_{i,r}, d_{i,r}, z_{i,r}, x_{i,r} )$ and the true values are estimated on a sample with $N = 100,000$.

§.§.§ Effects of interest and estimators

To compare the $\Delta$CBGATE with the $\Delta$GATE, consider both effects:

\begin{align}
\theta^{\Delta G} &= \E \left[Y_{1,1} - Y_{0,1} | Z = 1\right] - \E\left[Y_{1,0} - Y_{0,0} | Z = 0\right] \label{GATE} \\
\theta^{\Delta C} &= \E \left[Y_{1,1} - Y_{0,1} - Y_{1,0} + Y_{0,0}\right] \label{MGATE}
\end{align}

These two effects are only identical if there is no correlation between the moderator $Z$ and the covariates $X$. We estimate Equation <ref> and <ref> with DML with 5-fold cross-fitting in the following versions:

Simulation study: Versions of the DML estimators

$\theta^{\Delta C}$      $\theta^{\Delta G}$
propensity score      propensity score
(1)      $P(D = d, Z = z | X )$     $P(D = d | X)$

(2)      $P(D = d| Z, X)$, $P(Z = z | X)$

* Note: This table depicts different versions of DML estimators used for $\theta^{\Delta C}$. We use either the two marginal propensity scores or the joint propensity score. All weights have been normalised using Algorithm <ref>.

Random forests (number of trees: $1,000$) are used to estimate the nuisance functions. Algorithm <ref> in Section <ref> shows the $\Delta$CBGATE, while Algorithm <ref> in Appendix <ref> summarises the implemented GATE estimator. The GATE is estimated by separately estimating an ATE in the groups $Z = 1$ and $Z = 0$. To obtain $\theta^{\Delta G}$, the difference between those two ATEs is taken.

§.§.§ Simulation design

In total, four different simulation settings are estimated. The first element we change across settings are the numbers of replications $R$ and observations $N$. We run two simulations with $N = 2,500$ and $R = 1,000$ and two with $N = 10,000$ and $R = 250$. Furthermore, as explained above, we vary the correlation of $X$ and $Z$.
The optimal hyperparameters for the random forests, namely the maximum tree depth and the minimum leaf size, are again tuned by a grid search (maximum tree depth: [5, 10, 15, 20], minimum leaf size: [2, 5, 10, 20]). They are tuned using five different data draws and then fixed for the whole simulation. The optimal combination of maximal tree depth and minimum leaf size is chosen by taking the combination that appears most often in the five draws. The optimal parameters are shown in Table <ref>.

Simulation Study: Optimal hyperparameters for random forests ($\Delta$CBGATE)

max width=

13cDGP: correlation between moderator $Z$ and some covariates $X$
13cN = 2,500
$\mu_1$     $\mu_0$     $\mu_{11}$     $\mu_{01}$     $\mu_{10}$     $\mu_{00}$     $\pi_d$     $\lambda_z$     $\omega_{11}$     $\omega_{01}$     $\omega_{10}$     $\omega_{00}$

Number of trees     1,000     1,000     1,000     1,000    1,000    1,000     1,000     1,000     1,000     1,000    1,000    1,000

Maximal tree depth     10     5     5     5     10     5     5     5     5     5     5     5

Minimum leaf size     2     5     2     2     10     20     2     20     20     20     10     20
13cN = 10,000
$\mu_1$     $\mu_0$     $\mu_{11}$     $\mu_{01}$     $\mu_{10}$     $\mu_{00}$     $\pi_d$     $\lambda_z$     $\omega_{11}$     $\omega_{01}$     $\omega_{10}$     $\omega_{00}$

Number of trees     1,000     1,000     1,000     1,000    1,000    1,000     1,000     1,000     1,000     1,000    1,000    1,000

Maximal tree depth     5     5     5     10     5     5     10     5     5     5     5     5

Minimum leaf size     10     2     2     20     20     20     20     2     20     20     5     20
13cDGP: no correlation between moderator $Z$ and some covariates $X$
13cN = 2,500
$\mu_1$     $\mu_0$     $\mu_{11}$     $\mu_{01}$     $\mu_{10}$     $\mu_{00}$     $\pi_d$     $\lambda_z$     $\omega_{11}$     $\omega_{01}$     $\omega_{10}$     $\omega_{00}$

Number of trees     1,000     1,000     1,000     1,000    1,000    1,000     1,000     1,000     1,000     1,000    1,000    1,000

Maximal tree depth     10     5     10     5     10     5     5     5     5     5     5     5

Minimum leaf size     5     10     5     20     10     20     20     20     20     20     20     10
13cN = 10,000
$\mu_1$     $\mu_0$     $\mu_{11}$     $\mu_{01}$     $\mu_{10}$     $\mu_{00}$     $\pi_d$     $\lambda_z$     $\omega_{11}$     $\omega_{01}$     $\omega_{10}$     $\omega_{00}$

Number of trees     1,000     1,000     1,000     1,000    1,000    1,000     1,000     1,000     1,000     1,000    1,000    1,000

Maximal tree depth     10     5     10     5     10     5     5     5     5     5     5     5

Minimum leaf size     5     10     10     20     20     5     2     10     20     20     20     20

* Note: This table depicts the optimal hyperparameters for the random forests. The grid values are as follows: maximum tree depth: [5, 10, 15, 20], minimum leaf size: [2, 5, 10, 20].

§.§.§ Results

Figures <ref> - <ref> compare the distribution of the biases for $\hat{\theta}^{\Delta C}$ and $\hat{\theta}^{\Delta G}$ to a normal distribution. The bias of both estimators is calculated using the true $\theta^{\Delta C}$. The first finding is that the $\hat{\theta}^{\Delta G}$ estimator can be substantially biased if $Z$ and $X$ are correlated. This bias can be seen in Figure <ref> and  <ref>. In addition, comparing the two different sample sizes shows that if $N$ increases from 2,500 to 10,000, the standard error halves. Looking at Figure <ref> and <ref>, the estimators of $\hat{\theta}^{\Delta C}$ and $\hat{\theta}^{\Delta G}$ are very similar when there is no correlation between  $Z$ and $X$. Moreover, their distributions appear to converge to a normal distribution.

N $= 2,500$, correlated $Z$ and $X$

N = 2500, uncorrelated $Z$ and $X$

N = 10,000, correlated $Z$ and $X$

N = 10,000, uncorrelated $Z$ and $X$

Note: These figures show the distribution of the bias of $\hat{\theta}^{\Delta C}$ and $\hat{\theta}^{\Delta G}$ compared to the true $\theta^{\Delta C}$. We plot a normal distribution to indicate convergence to an appropriately normal distribution. The figures are created using the results of the unconfoundedness setting and the DML estimator with the joint propensity score estimator.

Table <ref> shows the results for $N = 2,500$ and $R = 1,000$. The first block presents the results for the setting where there is selection into treatment and correlation between $Z$ and $X$. The second block depicts the results without correlating $Z$ and $X$. Estimating the effect ($\hat{\theta}^{\Delta C}$) with a joint propensity score (Specification 1) or with the product of two marginal propensity scores (Specification 2) leads to qualitatively similar results.

Simulation results: N = 2,500

max width=

11cNumber of observations = 2,500
6lEstimation of effects    3lEstimation of standard errors
Spf.     Truth     Bias     |Bias|     Std     RMSE     Skew.     Ex. Kurt.    Bias (SE)     CovP (95)     CovP (80)

(in %)     (in %)
11cCorrelation between $Z$ and $X$
CBGATE_correlation_Yes_results_nonlinear_2500_1000.tex    11cNo correlation between $Z$ and $X$

* Note: This table shows results for $N = 2,500$ and $R = 1,000$ in two blocks: (1) correlation between $Z$ and $X$, and (2) no correlation between $Z$ and $X$. Column (1) shows the effect estimated, column (2) the respective version of the DML estimator used as explained in Table <ref> (3) the true effect. The remaining ten columns depict performance measures explained in <ref>.

Table <ref> shows results for $N = 10,000$ and $R = 250$. Compared to the results with the smaller sample presented in Table <ref>, the standard errors and the RMSE halve. This aligns with the convergence rate of a $\sqrt{N}$-consistent estimator.

Simulation results: N = 10,000

max width=

11cNumber of observations = 10,000
6lEstimation of effects    3lEstimation of standard errors
Spf.     Truth     Bias     |Bias|     Std     RMSE     Skew.     Ex. Kurt.    Bias (SE)     CovP (95)     CovP (80)

(in %)     (in %)
11cCorrelation between $Z$ and $X$
CBGATE_correlation_Yes_results_nonlinear_10000_250.tex    11cNo correlation between $Z$ and $X$

* Note: This table shows results for $N = 10,500$ and $R = 250$ in two blocks: (1) correlation between $Z$ and $X$, and (2) no correlation between $Z$ and $X$. Column (1) shows the effect estimated, column (2) the respective version of the DML estimator used as explained in Table <ref> (3) the true effect. The remaining ten columns depict performance measures explained in <ref>.

§ APPENDIX: EMPIRICAL EXAMPLE

§.§ Data descriptives

Table <ref> shows the mean and standard deviation of the covariates included in the analyses by treatment status. In addition to the covariates in this table, we add caseworkers' fixed effects. For a more detailed description of the data and how the dataset was constructed, please see <cit.>. The data can be accessed on swissubase.ch for research purposes.

Empirical analysis: Balance table for treatment and moderator variable

max width=0.95

Treated     Treated    Non-treated     Non-treated
Non-Swiss     Swiss    Non-Swiss     Swiss
Variable    Mean    Mean     Mean    Mean
descriptives_both.tex Number of observations     4438     8607    30417      47877

* Note: This table shows the mean of some covariates included in the analysis. Column (1) and (2) show it for treated individuals, column (3) and (4) for non-treated individuals. Column (1) and (3) show it for non-Swiss individuals, column (2) and (4) for Swiss individuals.

Table <ref> shows the mean and standard deviation of the covariates included in the analyses by treatment status. In addition to the covariates in this table, we add caseworkers' fixed effects. For a more detailed description of the data and how the dataset was constructed, please see <cit.>. The data can be accessed on swissubase.ch for research purposes.

Empirical analysis: Balance table for treatment variable (participation in the programme)

max width=0.95

2cTreated     2cControl

Covariates     Mean     Std. Dev.      Mean     Std. Dev.     Std. Diff.
descriptives.tex Number of observations     2c13,045     2c78,294

* Note: This table shows the mean and standard deviation of the covariates included in the analyses. Columns (1) and (2) show it for the treated group, columns (3) and (4) for the control group. The last column shows the standardised difference between the two groups. The standardized difference is calculated as $ SD=\frac{\left|\bar{X}_\text{treated}-\bar{X}_\text{control}\right|}{\sqrt{1 / 2\left(\operatorname{Var}\left(\bar{X}_\text{treated}\right)+\operatorname{Var}\left(\bar{X}_\text{control}\right)\right)}} \cdot 100$ where $\bar{X}_\text{treated}$ and $\bar{X}_\text{control}$ indicate the sample mean of the treatment and control group, respectively.

Table <ref> shows descriptive statistics for Swiss and non-Swiss individuals for the other covariates. It can be seen that there are some differences in the covariates, namely in the mother tongue, past income, the previous job and the qualifications. Other variables such as age, place of residence or caseworker characteristics are well balanced.

Empirical analysis: Balance table for moderator variable (nationality)

max width=

2cNon Swiss     2cSwiss

Covariates     Mean     Std. Dev.      Mean     Std. Dev.     Std. Diff.
descriptives_moderator.tex Number of observations     2c34,855     2c56,484

* Note: This table shows the mean and standard deviation of some covariates included in the analysis. Column (1) and (2) show it for Swiss individuals, column (3) and (4) for non-Swiss individuals. The last column shows the standardised difference between the two groups. The standardized difference is calculated as $ SD=\frac{\left|\bar{X}_\text{Swiss}-\bar{X}_\text{non-Swiss}\right|}{\sqrt{1 / 2\left(\operatorname{Var}\left(\bar{X}_\text{Swiss}\right)+\operatorname{Var}\left(\bar{X}_\text{non-Swiss}\right)\right)}} \cdot 100$ where $\bar{X}_\text{Swiss}$ and $\bar{X}_\text{non-Swiss}$ indicate the sample mean of the Non-Swiss and the Swiss individuals, respectively.

get arXiv to do 4 passes: Label(s) may have changed. Rerun


# Physics over a finite field and Wick rotation

B.Zilber

###### Abstract

The paper develops an earlier proposition that the physical universe is a
finite system co-ordinatised by a huge finite field
$\mathrm{F}_{\mathfrak{p}}$ which looks like the field of complex numbers to
an observer.

We construct a place (homomorphism) $\mathrm{lm}$ from a pseudo-finite field
$\mathrm{F}_{\mathfrak{p}}$ onto the compactified field of complex numbers in
such a way that certain multiplicative subgroups
${}^{\prime}{\mathbb{R}}^{\prime}_{+}$ and ${}^{\prime}\mathbb{S}^{\prime}$
correspond to the polar coordinate system ${\mathbb{R}}_{+}\times\mathbb{S}$
of ${\mathbb{C}}.$ Thus $\mathrm{F}_{\mathfrak{p}},$
${}^{\prime}{\mathbb{R}}^{\prime}_{+}$ and ${}^{\prime}\mathbb{S}^{\prime}$
provide co-ordinates for physical universe.

We introduce natural units for ${}^{\prime}{\mathbb{R}}^{\prime}_{+}$ and
${}^{\prime}\mathbb{S}^{\prime}$ and show that the passage from the scale of
units in ${}^{\prime}{\mathbb{R}}^{\prime}_{+}$ to the scale of units of
${}^{\prime}\mathbb{S}^{\prime}$ corresponds to a multiplication (on the
logarithmic scale) by a ’huge’ (non-standard) integer $\mathfrak{i}$ equal
approximately to $\sqrt{\mathfrak{p}}.$ This provides an explanation to the
phenomenon of Wick rotation.

In the same model we explain the phenomenon of phase transition in a large
finite system

##  1 Introduction

###### 1.1

The hypothesis that the universe is infinite is an open question. This
concerns both the size of the universe and the number of atoms or elements
that comprise it. Since it is now accepted that there is a minimal length, the
Planck length, assumption of the spatial finiteness of the universe implies
the assumption of the finiteness of the number of its elements.

In [1] we discussed the concept of approximation in physics and the suggestion
that physics universe is co-ordinatised by a huge111We use huge for numbers
which some physics authors call “ridiculously large”, see the discussion in
2.6 below. finite field $\mathrm{F}_{\mathfrak{p}}.$ It was proved
(Proposition 5.2 of [1]) that the only metric field (locally compact field)
that can be approximated by finite fields is the field ${\mathbb{C}}$ of
complex numbers. Thus “seen from afar” the huge finite field looks like a
field of complex numbers ${\mathbb{C}}.$

The current work has been inspired by Hao Hu, the expert in Philosphy of
Physics, who approached the author with the suggestion to apply the idea of
[1] to statistical mechanics and to attack the well-discussed problem of phase
transition (which happen in large finite systems but require the assumption of
infinity for its mathematical theory). We give here an answer to this problem.
Another important outcome of the mathematical theory developed below is the
interpretation of the phenomenon of Wick rotation as the result of the change
of scales in physics.

I would also like to note that a compatible attempt to develop the
mathematical background for a theory of finite physics was presented in [2],
[3] and [4].

###### 1.2

The main mathematical problem in making a practical use of the idea of a
“physics over a finite field $\mathrm{F}_{\mathfrak{p}}$” is to find a way of
representing the common-sense real quantities of physics inside the huge
finite field $\mathrm{F}_{\mathfrak{p}}$ and to find such a representation
that allows the standard physics calculations. That is to explain how both
${\mathbb{R}}$ and $i{\mathbb{R}}$ emerge from a large finite field.

In this regard it is useful to invoke a notion of feasible numbers that was
discussed by philosphers of mathematics, mathematical logicians and computer
scientists, see e.g. [8] for a mathematical treatment of the notion. Roughly
speaking, $1,2,\ldots,1000,$ as well as $1/5$ and $0.203$ are feasible
numbers, but the number of atoms in the visible universe, estimated to be
around $10^{80},$ is not feasible.

Thus, while we work with feasible numbers $1,2,\ldots$ inside
$\mathrm{F}_{\mathfrak{p}}$ we can think on these as the usual integers, but
when our integers become much much bigger then, according to the approximation
theorem of [1], they start to behave like complex numbers, e.g. if an integer
$\mathfrak{i}$ satisfies $\mathfrak{i}^{2}+1=0\mod\mathfrak{p}$ it takes the
role of $\sqrt{-1}.$

Thus the problem becomes to develop a mathematically useful equivalent of
feasible numbers along with a notion of a “huge prime” $\mathfrak{p}$ that
gives rise to the “huge finite field” $\mathrm{F}_{\mathfrak{p}}.$

###### 1.3

In the current work we follow the model theoretic motivation of [1] and
interpret a “huge finite field” as a pseudo-finite field
$\mathrm{F}_{\mathfrak{p}}$ which can be obtained by considering a non-
principal ultrafilter $\mathcal{D}$ on the set of prime numbers
$P\subset{\mathbb{Z}}_{+}$ (positive integers) and the ultraproduct

$\mathrm{F}_{\mathfrak{p}}=\prod_{p\in P}\mathrm{F}_{p}/\mathcal{D}.$

Model theory tells us that $\mathrm{F}_{\mathfrak{p}}$ can equaivalently be
obtained as the quotient-ring of the ring ${{}^{*}\mathbb{Z}}$ of non-standard
integers by the prime ideal $\mathfrak{p}\,{{}^{*}\mathbb{Z}},$ where
$\mathfrak{p}$ is the respective non-standard prime number,

$\mathrm{F}_{\mathfrak{p}}\cong{{}^{*}\mathbb{Z}}/\mathfrak{p}\mbox{ where
}{{}^{*}\mathbb{Z}}={\mathbb{Z}}^{P}/\mathcal{D}.$

Consequently, we adapt a simplfied interpretation of “feasible” and think of
standard integers  in ${{}^{*}\mathbb{Z}}$ as feasible.

However, along with such feasible numbers we are also interested in “very
large” numbers, interpreted as some non-standard numbers which are much
smaller than $\mathfrak{p}.$ An important example of such a number is
$\mathfrak{l}$ below.

###### 1.4

Along with $\mathfrak{p}$ and the field
$(\mathrm{F}_{\mathfrak{p}};+,\cdot,0,1)$ we specify:

\- a non-standard higly divisible number $\mathfrak{l}$ (each standard integer
$m$ divides $\mathfrak{l}$) satisfying some other assumptions below;

\- a two-sorted pseudo-finite structure
$({\mathbb{U}}_{\mathfrak{p},\mathfrak{l}},\mathrm{F}_{\mathfrak{p}})$ with

${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}=({{}^{*}\mathbb{Z}}/_{(\mathfrak{p}-1)\mathfrak{l}};+,\hat{0},\hat{1})$

a pseudo-finite additive cyclic222here and below “cyclic” means in the pseudo-
finite sense, i.e. the ultraproduct of cyclic groups  group of order
$(\mathfrak{p}-1)\mathfrak{l}$ with generator $\hat{1};$

\- a surjective group homomorphism

$\exp_{\mathfrak{p}}:{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}\to\mathrm{F}_{\mathfrak{p}}^{\times};\
\ \ n\cdot\hat{1}\mapsto\epsilon^{n}$

where $\epsilon$ is a generator of the (pseudo)-cyclic group
$\mathrm{F}_{\mathfrak{p}}^{\times},$ $n\in{{}^{*}\mathbb{Z}}.$ It follows
that

$\ker\exp_{\mathfrak{p}}=(\mathfrak{p}-1)\cdot{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$

the subgroup generated by $(\mathfrak{p}-1)\cdot\hat{1}$ (and suggests that
$\mathfrak{p}-1$ of ${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$ should be
interpreted as $2\pi i$);

\- a pair of surjective “limit” homomorphisms (places) $\mathrm{lm}$ which
make the diagram commute

$\begin{array}[]{lllll}\mathrm{lm}:\ \ \
{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}\ \ \ \ \twoheadrightarrow\ \ \ \ \
{\mathbb{C}}\cup\\{\infty\\}\\\ \\\ \ \ \ \exp_{\mathfrak{p}}\downarrow\ \ \ \
\ \ \ \ \ \ \ \ \ \exp\downarrow\\\ \\\ \mathrm{lm}:\ \ \
\mathrm{F}_{\mathfrak{p}}\ \ \ \ \to\ \ \ \ \
{\mathbb{C}}\cup\\{\infty\\}\end{array};$ (1)

There is a natural cyclic order on ${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$
in which $u+\hat{1}>u$ and there is a related cyclic order on
$\mathrm{F}_{\mathfrak{p}}^{\times}$ in which $\epsilon^{n+1}>\epsilon^{n}$
for all $n\in{{}^{*}\mathbb{Z}}.$

We treat $\hat{1}$ as an infinitesimal and choose two “units of length”
$\mathbf{u}$ and $\mathbf{v},$ elements of
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}},$

$\hat{1}<<\mathbf{u}<<\mathbf{v}.$

More precisely,

$\mathbf{u}=\frac{\mathfrak{p}-1}{\mathfrak{i}}\mbox{ and
}\mathbf{v}=\mathfrak{p}-1$ (2)

for some $\mathfrak{i}\in{{}^{*}\mathbb{Z}}_{+}$ such that

$\mathfrak{i}\mathfrak{l}|(\mathfrak{p}-1),\ \ \mathfrak{i}>>\mathfrak{l}$ (3)

($\mathfrak{i}\mathfrak{l}$ divides $\mathfrak{p}-1$).

Thus, $\mathbf{u}$ and $\mathbf{v}$ are units of two very different scales
(see 1.6 below for further comment).

We assume that

$\mathfrak{i}=\iota^{2}\mbox{ and }\mathfrak{l}=\mu^{2}$ (4)

for some $\mu,\iota\in{{}^{*}\mathbb{Z}}_{+}.$

We also need to assume

$\mathfrak{i}^{2}+1=\mathfrak{p}\mbox{ \ \ or\ \ \ $\mathfrak{i},\mathfrak{l}$
algebraically independent in }\mathrm{F}_{\mathfrak{p}}$ (5)

(the first is the preferable and more elegant assumption but it is not known
whether it is consistent with $\mathfrak{p}$ being infinite (the Landau
problem)).

It is easy to check that our set of assumptions (3)-(5) along with the
assumption of high divisibility of $\mathfrak{l}$ are consistent. We are going
to slightly extend these assumption later, in particular (22) assumes that
$\mathfrak{l}^{n}<\mathfrak{p}$ for all $n\in{\mathbb{Z}}_{+}.$

We also define additive subgroups of
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$ called suggestively
${}^{\prime}{\mathbb{R}}^{\prime}$ and ${}^{\prime}i{\mathbb{R}}^{\prime}.$

###### 1.5

Main Theorem. There exists a surjective ring homomorphism (place)

$\mathrm{lm}_{\mathrm{F}}:\mathrm{F}_{\mathfrak{p}}\twoheadrightarrow{\mathbb{C}}\cup\\{\infty\\}$

and a surjective additive semigroup homomorphism

$\mathrm{lm}_{\mathbb{U}}:{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}\twoheadrightarrow{\mathbb{C}}\cup\\{\infty\\}$

such that:

$\exp\circ\mathrm{lm}_{\mathbb{U}}=\mathrm{lm}_{\mathrm{F}}\circ\exp_{\mathfrak{p}};$
(6)
$\mathrm{lm}_{\mathbb{U}}:{{}^{\prime}{\mathbb{R}}^{\prime}}\twoheadrightarrow{\mathbb{R}};$
(7)
$\mathrm{lm}_{\mathbb{U}}:{{}^{\prime}i{\mathbb{R}}^{\prime}}\twoheadrightarrow
i{\mathbb{R}};$ (8)

for any $l\in{{}^{*}\mathbb{Z}}$ such that $|l|<m\mathfrak{l}$ for some
standard integer $m$:

$\mathrm{lm}_{\mathrm{F}}:\exp_{\mathfrak{p}}(l\mathbf{u})\mapsto e^{-\pi z},\
\ \mbox{ for }z:=\mathrm{st}(\frac{l}{2\mathfrak{l}})\in{\mathbb{R}}$ (9)

$\mathrm{lm}_{\mathrm{F}}:\exp_{\mathfrak{p}}(l\mathbf{v})\mapsto e^{i\pi y},\
\ \mbox{ for }y:=\mathrm{st}(\frac{l}{2\mathfrak{l}})\in{\mathbb{R}}$ (10)

where $\mathrm{st}$ is the standard part map.

For $a\in{\mathbb{Q}}_{+},$ a complete square and $\mu\in{{}^{*}\mathbb{Z}}$
such that $\mu^{2}=\mathfrak{l}:$

$\mathrm{lm}_{\mathrm{F}}:\frac{1}{\mu}\mapsto 0$ (11)
$\mathrm{lm}_{\mathrm{F}}:\ \frac{1}{\mu}\sum_{-\mathfrak{i}\mathfrak{l}/a\leq
n<\mathfrak{i}\mathfrak{l}/a}\exp_{\mathfrak{p}}(a\frac{n^{2}}{2\mathfrak{l}}\mathbf{u})\
\ \mapsto\ \ \int_{\mathbb{R}}e^{-a\pi x^{2}}dx$ (12)

where $\int_{\mathbb{R}}e^{-ax^{2}}dx=\frac{1}{\sqrt{a}}$ according to the
standard definition of the Gaussian integral.

$\mathrm{lm}_{\mathrm{F}}:\ \frac{1}{\mu}\sum_{-\mathfrak{l}/a\leq
n<\mathfrak{l}/a}\exp_{\mathfrak{p}}(a\frac{n^{2}}{2\mathfrak{l}}\mathbf{v})\
\ \mapsto\ \ \int_{\mathbb{R}}e^{ia\pi x^{2}}dx$ (13)

where $\int_{\mathbb{R}}e^{ia\pi x^{2}}dx=\frac{e^{\frac{\pi
i}{4}}}{\sqrt{a}}$ according to the Quantum Mechanics calculus.

Note that (7) and (8) give us subgroups of
$\mathrm{F}_{\mathfrak{p}}^{\times}$

${{}^{\prime}{\mathbb{R}}^{\prime}}_{+}:=\exp_{\mathfrak{p}}(^{\prime}{\mathbb{R}}^{\prime})\mbox{
and
}{{}^{\prime}\mathbb{S}^{\prime}}:=\exp_{\mathfrak{p}}(^{\prime}i{\mathbb{R}}^{\prime})$

which together with (6) furnish a good analogue of polar coordinate system in
$\mathrm{F}_{\mathfrak{p}}.$

###### 1.6

Discussion. The theorem clarifies the relationship between two different
scales in
$({\mathbb{U}}_{\mathfrak{p},\mathfrak{l}},\mathrm{F}_{\mathfrak{p}})$
presented by units $\mathbf{u}$ and $\mathbf{v}.$ These should be though of as
units for physics of ’low energy’ and ’high energy’, respectively, the latter
being physics at quantum level and the former the physics at the level of
Brownian motion.

The statements (9) and (10) demonstrate that the action of the huge
pseudofinite integer $\mathfrak{i}$ which changes the scale of units in
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$ (recall that
$\mathbf{v}=\mathfrak{i}\cdot\mathbf{u}$, and
$\mathfrak{i}\approx\sqrt{\mathfrak{p}}$) is seen in ${\mathbb{C}}$ as the
Wick rotation $e^{\pi z}\mapsto e^{i\pi z}.$

The Gaussian integrals in (12) and (13) are mathematical manifestation of the
same phenomena expressed by the summation formulae over
$\mathrm{F}_{\mathfrak{p}}$ on the left hand side of $\mapsto.$ The difference
in the integral expressions on the right comes from the difference in the
scale of units that measure the respective processes.

Thus the puzzling regularity of the transition from the Brownian motion
integral (12) to the quantum mechanics integral (13) known and exploited by
physicists as Wick rotation has an explanation as a mathematical consequence
of the change of scales.

###### 1.7

Discussion. The “limit” $\mathrm{lm}_{\mathrm{F}}$ maps the discrete field
$\mathrm{F}_{\mathfrak{p}}$ into (the compactification of) the field of
complex numbers and thus endows the image with a metric. The Theorem
determines $\mathrm{lm}_{\mathrm{F}}$ on the subring generated by specific
points (see (9),(10) and (11)) but leaves the rest free. This means that the
observer, which sees $\mathrm{F}_{\mathfrak{p}}$ with all the algebraic
geometry over it through $\mathrm{lm}_{\mathrm{F}},$ has some freedom in
choosing the metric on algebraic varieties. In particular, if
$Z\subseteq\mathrm{F}_{\mathfrak{p}}^{m}$ is an algebraic subvariety, say a
torus, then $\mathrm{lm}_{\mathrm{F}}(Z)$ is a subset of the compactification
of ${\mathbb{C}}^{m},$ a compact complex variety. In general, such a
compactification is far of being unique.

Thus the freedom in the choice of $\mathrm{lm}_{\mathrm{F}}$ implies a
respective degree of freedom in the choice of complex/metric version of
physics.

##  2  Statistical physics and phase transition

###### 2.1

Physical units and dimensions

In the formalism of two sorts ${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$ and
$\mathrm{F}_{\mathfrak{p}}$ introduced in 1.4 the sort
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$ is assumed to be the sort thar
holds all the physical units (dimensions). It is convenient for each principal
unit of measurement to define a special sort $\mathbb{D}_{i},$ $i=1,\ldots,k$
which is going to be naturally interpreted in terms of
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}.$ .

Each sort $\mathbb{D}_{i}$ is a subgroup of
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$ and so has a cyclic additive group
structure with the unit (generator)
$\mathbf{d}_{i}\in{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$
($\mathbb{D}_{i}$-unit). $\mathbb{D}_{i}$ is isomorphic to
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}/\ker_{i},$ where

$\ker_{i}=\frac{(\mathfrak{p}-1)\mathfrak{l}}{\mathbf{d}_{i}}\cdot{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}},\
\ \mathbf{d}_{i}|\,(\mathfrak{p}-1)\mathfrak{l}.$

Thus the size of $\mathbb{D}_{i},$

$|\mathbb{D}_{i}|=\frac{(\mathfrak{p}-1)\mathfrak{l}}{\mathbf{d}_{i}}.$

Between some of the unit sorts there are bilinear maps

$\mathbb{D}_{1}\times\mathbb{D}_{2}\twoheadrightarrow\mathbb{D}_{3};\ \ \
(x_{1}\mathbf{d}_{1},x_{2}\mathbf{d}_{2})\mapsto x_{1}x_{2}\mathbf{d}_{3}$

where we assume $\ker_{3}=\ker_{1}\cap\ker_{2}$ and

$x_{1}=u_{1}+\ker_{1},\ x_{2}=u_{1}+\ker_{2},\ \ \ x_{1}\cdot
x_{2}:=u_{1}\cdot u_{2}+\ker_{3}$

in the ring ${{}^{*}\mathbb{Z}}/(\mathfrak{p}-1)\mathfrak{l}.$

Thus counting in units of $\mathbb{D}_{3}$ gives

$x_{3}=x_{1}x_{2}$

and this can be equivalently written as

$x_{1}=x_{2}^{-1}x_{3}.$

The units $\mathbf{u}$ and $\mathbf{v}$ introduced in 1.4 are examples of
$\mathbb{D}$-units.

###### 2.2

The field sort $\mathrm{F}_{\mathfrak{p}}$ is assumed to be dimensionless and
the exponentiation map $\exp_{\mathfrak{p}}$ restricted to a sort
$\mathbb{D}_{i}$ is a homomorphism

$\exp_{\mathfrak{p}}:\mathbb{D}_{i}\twoheadrightarrow\mathrm{F}_{\mathfrak{p}}^{\times};\
\ n\cdot\mathbf{d}_{i}\mapsto\exp_{\mathfrak{p}}(n\mathbf{d}_{i}).$

###### 2.3

In Statistical Mechanics the dimensions in
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$ are usually Energy (E), Temperature
(T) and (in ferromagnets) Magnetisation (H).

According to this theory probability that the system in temperature $T$ is in
a state $\sigma$ is equal to $\exp(-\frac{E_{\sigma}}{kT},$ and the
probability $p_{n}$ that the system consists of exactly $n$ atoms is

$P_{n}=\sum_{\sigma\in\Sigma(n)}\exp(-\frac{E_{\sigma}}{kT}),$

where $\Sigma(n)$ are all the states with exactly $n$ atoms.

Setting (with some simplifications) $y:=\exp(-\frac{H}{kT}),$ the equilibrium
state of the system of volume $N$ (that is having up to $N$ particles) is
analysed via the polynomial

$\mathcal{P}_{N}(y):=\sum_{n=0}^{N}P_{n}y^{n}.$

which is called the grand partition function of the system.

In effect, $P_{n}=\frac{p_{n}}{Z}$ where $Z$ is the number of all possible
states and $p_{n}$ the number of states with $n$ particles, both integers.
Since there are very few restrictions on states in the models the number $Z$
is close to the number of all possible subsets, that is

$\sum_{n=0}^{N}p_{n}=Z\approx 2^{N}.$ (14)

We will assume, without loss of generality, that $P_{n}=p_{n},$ that is that
the coefficients of $\mathcal{P}_{N}(y)$ are positive integers.

###### 2.4

The seminal work of C.N.Yang and T.D.Lee, [5] \- [6] (1952), laid the ground
for the modern theory of critical points in the evolution of large finite
systems such as the ideal gas.

The main theorem of [5] states that the phase transition in the system happens
at the point $y_{\mathrm{crit}}$ (a critical point) such that

$\mathcal{P}_{N}(y_{\mathrm{crit}})=0$

which obviosly can not be a real point. The paper analyses complex roots of
the polynomial which proves very important for the behaviour of the system
near the critical point.

The modern theory resolves this paradox by assuming $N\to\infty,$ in which
case $\mathcal{P}_{N}(y)$ converges to an analytic function and
$y_{\mathrm{crit}}$ converges to a real point (the passage to the
thermodynamical limit). This solution of the paradox is not considered to be
satisfactory as the actual systems are always finite, although very large.

###### 2.5

Our, alternative solution, uses the assumption of the universe over finite
field. Under this assumption $y_{\mathrm{crit}}\in\mathrm{F}_{\mathfrak{p}},$
that is $y_{\mathrm{crit}}$ is a non-standard integer such that

$\mathcal{P}_{N}(y_{\mathrm{crit}})=0\mod\mathfrak{p}.$ (15)

This is consistent but requires $y_{\mathrm{crit}}$ to be a very large non-
feasible integer.

One can explore the assumption further using the reduction (24) from [6]
together with Theorem 3 therein which states that after the reduction all the
zeroes of $\mathcal{P}_{N}$ are on the unit circle and the limit of the zeroes
is $=1.$ This leads us to the conclusion that

$\mathcal{P}_{N}(1)=0\mod\mathfrak{p}$

Equivalently,

$\sum_{n=1}^{N}p_{n}=0\mod\mathfrak{p}.$

Combining this with (14) and assuming that the number of atoms $N$ is $\approx
10^{23},$ the Avogadro number, we can make an estimate on $\mathfrak{p},$ the
upper bound:

$\mathfrak{p}<2^{N}\approx 2^{10^{23}}$ (16)

On the other hand the same argument proves that there is a low bound on the
volume $N$ of gas which allows a phase transition, that is has a critical
point satisfying (15):

$N>\log\mathfrak{p}.$

###### 2.6

Discussion. The assumption of physics over a finite field explains the
necessity of extending the definition of grand partition function as the
function of complex variable as well as explains the phase transition in a
large finite system.

The estimate (14) appears also in [7] on page 4 where the number is being
characterised as ridiculously large number.

The rest of the paper is purely mathematical. It provides the construction of
the limit map $\mathrm{lm}$ and proofs.

##  3 The pseudo-finite exponentiation

Fix notation for a non-standard model of ${\mathbb{C}}$

${{}^{*}{\mathbb{C}}}={\mathbb{C}}^{P}/\mathcal{D}$

where $P$ and the ultrafilter $\mathcal{D}$ on $P$ are defined in 1.3.

Note that by construction ${{}^{*}\mathbb{Z}}\subset{{}^{*}{\mathbb{C}}}$ and
this allows us to identify elements $l\in\mathrm{F}_{\mathfrak{p}}$ which are
represented by $l\in{{}^{*}\mathbb{Z}},$ $0\leq l<\mathfrak{p},$ with
$l\in{{}^{*}{\mathbb{C}}}$ in the theorem below.

###### 3.1

Theorem. Then there is a place
$\mathcal{I}:\mathrm{F}_{\mathfrak{p}}\to{{}^{*}{\mathbb{C}}}\cup\\{\infty\\}$
such that

$\mu\mapsto\mu\mbox{ and }\mathfrak{l}\mapsto\mathfrak{l}$ (17)

$\iota\mapsto e^{-\frac{\pi i}{4}}\mbox{ and }\mathfrak{i}\mapsto
e^{-\frac{\pi i}{2}}$ (18)

and for all $a\in{\mathbb{Q}},$ for all $M\in{{}^{*}\mathbb{Z}},$
$|M|<\mathfrak{l},$

$\epsilon^{\frac{aM(\mathfrak{p}-1)}{2\mathfrak{l}}}\mapsto e^{-\frac{aM\pi
i}{\mathfrak{l}}}$ (19)

$\epsilon^{\frac{aM(\mathfrak{p}-1)}{2\mathfrak{l}\mathfrak{i}}}\mapsto
e^{-\frac{aM\pi}{\mathfrak{l}}}.$ (20)

The proof is by Lemmata below.

We consider linear equations of the form

$\sum_{i=1}^{k}c_{i}\cdot X_{i}=1$

where the variables $X_{i}$ are assumed to be in a specific subset $G$ of the
field. A solution $x_{1},\ldots,x_{k}$ is said to be non-degenerate if for any
proper subset $K\subset\\{1,\ldots,k\\}$

$\sum_{i\in K}c_{i}\cdot x_{i}\neq 1.$

###### 3.2

Lemma. There is a function $f:{\mathbb{Z}}_{+}\to{\mathbb{Z}}_{+},$ a highly
divisible $\nu\in{{}^{*}\mathbb{Z}}_{+}$ and $\kappa\in{{}^{*}\mathbb{Z}}_{+}$
such that $2\nu\kappa|(\mathfrak{p}-1),$ for all $n\in{\mathbb{Z}}_{+}$
$\kappa>\nu^{n}$ and for all $k\in{\mathbb{Z}}_{+}$ for all rational functions
$c(X,Y)=\langle c_{1}(X,Y),\ldots,c_{k}(X,Y)\rangle$

$\left.\begin{array}[]{lll}\mbox{any non-degenerate solution
}x_{1},\ldots,x_{k}\in\mathrm{F}_{\mathfrak{p}}\\\ \mbox{of
}\sum_{i=1}^{k}c_{i}(\nu,\kappa)\cdot x_{i}=1\ \&\
\bigwedge_{i}x_{i}^{\nu}=1\\\ \mbox{satisfies
}\bigwedge_{i}x_{i}^{f(k)}=1.\end{array}\right\\}$ (21)

Proof. Recall that $\mathrm{F}_{\mathfrak{p}}$ is a field of characteristic 0.
Thus the well-known Theorem of Mann about linear equations in roots of unity
with rational coefficients is applicable. A consequence of Mann’s Theorem is
that there is a function $f$ satisfying (21) for any
$\nu,\kappa\in{\mathbb{Z}}_{+}$ (see [14] for this and other consequences).

We treat the expression $x^{\nu}$ as an arithmetic function of $x,\nu$ defined
in $({{}^{*}\mathbb{Z}};+,\cdot,\mathfrak{p})$ along with the interpretation
of the field $\mathrm{F}_{\mathfrak{p}}$.

Let $\mathcal{M}_{c}\subset{{}^{*}\mathbb{Z}}_{+}^{2}$ be the set of
$(\nu,\kappa)\in{{}^{*}\mathbb{Z}}_{+}^{2}$ such that (21) holds for given $k$
and $c(X,Y).$ Clearly $\mathcal{M}_{c}$ is definable in
$({{}^{*}\mathbb{Z}};+,\cdot,\mathfrak{p}).$ By the above consequence of the
Mann Theorem ${\mathbb{Z}}_{+}^{2}\subseteq\mathcal{M}_{c}$ and so

${\mathbb{Z}}_{+}^{2}\subseteq\bigcap_{c}\mathcal{M}_{c}$

where $c$ runs in all $k$-tuples of rational functions $c(X,Y).$

Since each $n\in{\mathbb{Z}}_{+}$ divides $\mathfrak{p}-1$ it follows that the
type

$\bigwedge_{c}(\nu,\kappa)\in\mathcal{M}_{c}\ \&\ 2\nu\kappa|(\mathfrak{p}-1)\
\&\ \bigwedge_{n\in{\mathbb{Z}}_{+}}n|\nu\ \&\ \kappa>\nu^{n}$

is consistent, thus has a realisation in the $\aleph_{0}$-saturated structure
${{}^{*}\mathbb{Z}}.$ $\Box$

###### 3.3

Corollary. We may assume that for all $k$ and $c(X,Y)$ (21) is satisfied when
$\nu:=\mathfrak{l}$ and, if $\mathfrak{i}^{2}+1\neq\mathfrak{p},$
$\kappa:=\mathfrak{i}$. In particular,

$\begin{array}[]{lll}\mathfrak{i}>\mathfrak{l}^{n},\mbox{ for all
}n\in{\mathbb{Z}}_{+}\\\ \mathfrak{l}\mbox{ transcendental over
${\mathbb{Q}}(\mathfrak{i})$ in }\mathrm{F}_{\mathfrak{p}}\\\
\mathfrak{i}^{2}+1=\mathfrak{p}\ \mbox{ or $\mathfrak{i}$ is transcendental in
}\mathrm{F}_{\mathfrak{p}}\par\end{array}$ (22)

###### 3.4

Set

${}^{\prime}i{\mathbb{R}}^{\prime}:=\\{{\frac{K}{m\mathfrak{l}}\mathbf{v}}:\
-l\mathfrak{l}/2\leq K\leq l\mathfrak{l}/2,\ l,m\in{\mathbb{Z}}_{+}\\}$

and let

${}^{\prime}\mathbb{S}^{\prime}:=\exp_{\mathfrak{p}}(^{\prime}i{\mathbb{R}}^{\prime}),$

a subgroup of all the elements of $\mathrm{F}^{\times}_{\mathfrak{p}}$ of
order $<m\mathfrak{l}$ for some $m.$

###### 3.5

Lemma. Let $\gamma_{1},\ldots,\gamma_{n}\in{{}^{\prime}\mathbb{S}^{\prime}}$
be multiplicatively independent. Then $\gamma_{1},\ldots,\gamma_{n}$ are
algebraically independent over ${\mathbb{Q}}(\mathfrak{l},\mathfrak{i}).$

Proof. Suppose not. Then for some $k,$ for some
$c_{1},\ldots,c_{k}\in{\mathbb{Q}}(\mathfrak{l},\mathfrak{i}),$ for some
monomials $x_{i}=x_{i}(\gamma_{1},\ldots,\gamma_{n}),$ $i=1,\ldots,k,$ the
equality $\sum_{i=1}^{k}c_{k}\cdot x_{i}=1$ holds. We assume that $k$ is
minimal with this property. Since $\gamma_{1},\ldots,\gamma_{n}$ are roots of
1 of order $<l\mathfrak{l}$ for some $l\in{\mathbb{Z}}_{+},$ so are their
products $x_{1},\ldots,x_{k-1}.$ Since $k$ is minimal, the solution
$x_{i}(\gamma_{1},\ldots,\gamma_{n}),$ $i=1,\ldots,k,$ of the equation is non-
degenerate. Thus the $x_{i}(\gamma_{1},\ldots,\gamma_{n})$ are roots of unity
of order $\leq f(k).$ This would contradict our assumption on multiplicative
independence. $\Box$

###### 3.6

Corollary. There is a place

$\mathcal{I}:{\mathbb{Q}}(\mathfrak{l},\mathfrak{i},{{}^{\prime}\mathbb{S}^{\prime}})\to{{}^{*}{\mathbb{C}}}\cup\\{\infty\\}$

which satisfies (17),(18) and (19).

Proof. Denote ${\mathbb{Q}}(\mathfrak{l},\mathfrak{i},\sqrt[\infty]{1})$ the
extension of ${\mathbb{Q}}(\mathfrak{l},\mathfrak{i})$ by roots of $1.$ A
place
$\mathcal{I}:{\mathbb{Q}}(\mathfrak{l},\mathfrak{i},\sqrt[\infty]{1}))\to{{}^{*}{\mathbb{C}}}\cup\\{\infty\\}$
satisfying (17) and (18) exists since $\mathfrak{l}$ is transcendental over
${\mathbb{Q}}(\mathfrak{i}),$ see (22). By Lemma 3.5 this can be extended to
${\mathbb{Q}}(\mathfrak{l},\mathfrak{i},{{}^{\prime}\mathbb{S}^{\prime}})\to{{}^{*}{\mathbb{C}}}$
$\Box$

Let

${}^{\prime}{\mathbb{R}}^{\prime}:=\\{{\frac{K}{m\mathfrak{l}}\mathbf{u}}:\
-l\mathfrak{l}/2\leq K\leq l\mathfrak{l}/2,\
l,m\in{\mathbb{Z}}_{+}\\}\subset{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}},$

a subgroup of ${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}},$

and

${}^{\prime}{\mathbb{R}}^{\prime}_{+}:=\exp_{\mathfrak{p}}(^{\prime}{\mathbb{R}}^{\prime})\subset\mathrm{F}_{\mathfrak{p}},$

a subgroup of $\mathrm{F}^{\times}_{\mathfrak{p}}.$

###### 3.7

Remark. Note that the definition of $\exp_{\mathfrak{p}}$ in 1.4 depends on
the choice of the generator $\epsilon.$ So the above
${}^{\prime}{\mathbb{R}}^{\prime}_{+}$ depends on $\epsilon$ as well (but
${}^{\prime}\mathbb{S}^{\prime}$ does not).

###### 3.8

Lemma. Let $k,m,l\in{\mathbb{Z}}_{+}$ and
$\alpha=\alpha(\epsilon,m)=\epsilon^{\frac{\mathfrak{p}-1}{m\mathfrak{l}\mathfrak{i}}}.$
Let $\bar{M}=\langle M_{1},\ldots,M_{k}\rangle$ in ${{}^{*}\mathbb{Z}}^{k},$
$|M_{i}|<l\mathfrak{l}.$

Let $\bar{g}(\bar{Z})=\langle
g_{1}(Z_{1},\ldots,Z_{k}),\ldots,g_{k}(Z_{1},\ldots,Z_{k})\rangle$ be a
$k$-tuple of rational functions over ${\mathbb{Q}}(\mathfrak{l},\mathfrak{i})$
and $\bar{c}=\langle c_{1},\ldots,c_{k}\rangle,$
$c_{i}\in{\mathbb{Q}}(\mathfrak{l},\mathfrak{i},{{}^{\prime}\mathbb{S}^{\prime}})$
of the form

$c_{i}=g_{i}(\bar{s}),\ \bar{s}:=\langle s_{1},\ldots,s_{k}\rangle\mbox{ for
}s_{1},\ldots,s_{k}\in{{}^{\prime}\mathbb{S}^{\prime}}(l,m).$

Consider a non-standard function in $\mathrm{F}_{\mathfrak{p}}$ of the form

$P_{\bar{c},\bar{M}}(X):=\sum_{i=1}^{k}c_{i}X^{M_{i}}-1.$ (23)

Then there is a generator $\epsilon$ of $\mathrm{F}_{\mathfrak{p}}$ such that
for all $\bar{M}$ and $\bar{s}$ as above

$P_{\bar{c},\bar{M}}(\alpha)\neq 0.$

Proof. For a given $\bar{M}$ and $\bar{c}$, $P_{\bar{c},\bar{M}}(X)$ has at
most $l\mathfrak{l}$ zeroes. There are at most $(lm\mathfrak{l})^{k}$ possible
tuples $\bar{c}=\bar{g}(\bar{s})$ and $(l\mathfrak{l})^{k}$ of tuples
$\bar{M},$ so at most $(lm\mathfrak{l})^{2k}$ polynomials (23) altogether, so
$(lm\mathfrak{l})^{2k+1}$ zeroes of the polynomials.

On the other hand $\alpha(\epsilon,m)$ takes any value of primitive root of
order $m\mathfrak{l}\mathfrak{i}$ as $\epsilon$ runs through generators of
$\mathrm{F}^{\times}_{\mathfrak{p}}.$ Thus there are
$\varphi(m\mathfrak{l}\mathfrak{i})$ (the Euler function) such $\alpha$ and
the well-known lower estimate gives us
$\varphi(m\mathfrak{l}\mathfrak{i})>\sqrt{\mathfrak{l}\mathfrak{i}},$ which is
bigger than $(lm\mathfrak{l})^{2k+1}$ by (22). Thus there is an $\epsilon$
such as $\alpha(\epsilon,m)$ is as required. $\Box$

###### 3.9

Corollary. There is $\epsilon$ such that $\alpha(\epsilon,m)$ is not a zero of
any $P_{\bar{c},\bar{M}}(X)$ for all $l,$ $m,$ $k,$ $\bar{M}$ and
$\bar{c}\in{\mathbb{Q}}(\mathfrak{l},\mathfrak{i},{{}^{\prime}\mathbb{S}^{\prime}})^{k}.$

Proof. Note that the product of finitely many of polynomials
$P_{\bar{c},\bar{M}}(X)$ is again a polynomial of the same form, so Lemma 3.8
implies that there is $\alpha$ which is not a zero of any finite number of
polynomials with coefficients from a finite set $\bar{g}^{*}(\bar{s}$ and the
degrees $M_{i}$ restricted by $l^{*}\mathfrak{l}$ for some fixed $l.$ This can
be restated as a formal statement
$\exists\epsilon\,\Phi_{m,l,\bar{g}}(\alpha(\epsilon,m),\mathfrak{l},\mathfrak{i},\mathfrak{p})$
in the language of arithmetic. Thus the type

$\\{\Phi_{m,l,\bar{g}}(\alpha(\epsilon,m),\mathfrak{l},\mathfrak{i},\mathfrak{p}):\
m,l,\bar{g}\\}$

in variable $\epsilon$ is consistent. Since ${{}^{*}\mathbb{Z}}$ is
$\aleph_{0}$-saturated there is an $\epsilon$ such that
$\alpha(\epsilon,m)\in\mathrm{F}_{\mathfrak{p}}$ is not a zero of any
polynomial (23) $\Box$

###### 3.10

Lemma. There is a generator $\epsilon$ such that any multiplicatively
independent
$\gamma_{1},\ldots,\gamma_{m}\in\,^{\prime}{\mathbb{R}}^{\prime}_{+}$ are
algebraically independent over
${\mathbb{Q}}(\mathfrak{l},\mathfrak{i},{{}^{\prime}\mathbb{S}^{\prime}}).$

Proof. Let $\epsilon$ be as stated in 3.9. Suppose
$\gamma_{1},\ldots,\gamma_{m}$ are multiplicatively independent and satisfy a
polynomial equation $R(\gamma_{1},\ldots,\gamma_{m})=0.$ The equation can be
rewritten as

$\sum_{j}c_{j}\gamma^{\bar{m}_{j}}=1$

for some monomials $\gamma^{\bar{m}_{j}}=\prod_{i}\gamma_{i}^{m_{ji}}$ and
$c_{j}\in{\mathbb{Q}}(\mathfrak{l},\mathfrak{i},{{}^{\prime}\mathbb{S}^{\prime}}).$

Clearly, $\gamma^{\bar{m}_{j}}$ belong to the group
${}^{\prime}{\mathbb{R}}^{\prime}_{+}$ and so
$\gamma^{\bar{m}_{j}}=\alpha(\epsilon,m)^{M_{j}},$ for some $m,$ $M_{j},$
$|M_{j}|<l\mathfrak{l},$ contradicting our choice of $\epsilon.$ $\Box$

###### 3.11

Corollary. The place $\mathcal{I}$ of 3.6 can be extended to

$\mathcal{I}:{\mathbb{Q}}(\mathfrak{l},\mathfrak{i},{{}^{\prime}\mathbb{S}^{\prime}},^{\prime}{\mathbb{R}}^{\prime}_{+})\to{{}^{*}{\mathbb{C}}}\cup\\{\infty\\}$

which satisfies (20).

Proof. $\mathcal{I}$ of (20) is an isomorphism of groups. Since the only
relations in the language of rings between elelements in
${}^{\prime}{\mathbb{R}}^{\prime}_{+}$ are multiplicative relations,
$\mathcal{I}$ is a place in the language of rings. $\Box$

###### 3.12

Since ${{}^{*}{\mathbb{C}}}$ is algebraically closed the $\mathcal{I}$ of 3.11
can be extended to a place

$\mathcal{I}:\mathrm{F}_{\mathfrak{p}}\to{{}^{*}{\mathbb{C}}}\cup\\{\infty\\}.$

This finishes the proof of Theorem 3.1.

$\Box$

###### 3.13

Theorem. There is a surjective place

$\mathrm{lm}_{\mathrm{F}}:\mathrm{F}_{\mathfrak{p}}\twoheadrightarrow{\mathbb{C}}\cup\\{\infty\\}$

and a surjective additive homomorphism

$\mathrm{lm}_{\mathbb{U}}:{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}\twoheadrightarrow{\mathbb{C}}\cup\\{\infty\\}$

such that:

$\exp\circ\mathrm{lm}_{\mathbb{U}}=\mathrm{lm}_{\mathrm{F}}\circ\exp_{\mathfrak{p}}$
$\mathrm{lm}_{\mathbb{U}}:\
{{}^{\prime}{\mathbb{R}}^{\prime}}\twoheadrightarrow{\mathbb{R}};\ \
{{}^{\prime}i{\mathbb{R}}^{\prime}}\twoheadrightarrow i{\mathbb{R}};$
$\mathrm{lm}_{\mathrm{F}}:\
{{}^{\prime}{\mathbb{R}}^{\prime}_{+}}\twoheadrightarrow{\mathbb{R}}_{+};\ \
{{}^{\prime}\mathbb{S}^{\prime}}\twoheadrightarrow\mathbb{S};$

$\mathrm{lm}_{\mathrm{F}}:\mu\to\infty;\ \ \iota\mapsto e^{-\frac{\pi i}{4}};$
(24)

for any $l\in{{}^{*}\mathbb{Z}}$ such that $|l|<m\mathfrak{l}$ for some
$m\in{\mathbb{Z}}_{+}$

$\mathrm{lm}_{\mathrm{F}}:\exp_{\mathfrak{p}}(l\mathbf{u})\mapsto e^{-\pi z},\
\ \mbox{ for }z:=\mathrm{st}(\frac{l}{2\mathfrak{l}})\in{\mathbb{R}}$ (25)

$\mathrm{lm}_{\mathrm{F}}:\exp_{\mathfrak{p}}(l\mathbf{v})\mapsto e^{-i\pi
z},\ \ \mbox{ for }z:=\mathrm{st}(\frac{l}{2\mathfrak{l}})\in{\mathbb{R}}$
(26)

where $\mathrm{st}$ is the standard part map.

Proof. Define

$\mathrm{lm}_{\mathrm{F}}:=\mathrm{st}\circ\mathcal{I}.$

In particular,

$\epsilon^{\frac{l}{2\mathfrak{l}}\mathbf{u}}\mapsto
e^{-\pi\,\mathrm{st}(\frac{l}{2\mathfrak{l}})}\mbox{ and
}\epsilon^{\frac{l}{2\mathfrak{l}}\mathbf{v}}\mapsto
e^{-i\pi\,\mathrm{st}(\frac{l}{2\mathfrak{l}})}.$

It is easy to check that

${{}^{\prime}{\mathbb{R}}^{\prime}}\cap{{}^{\prime}i{\mathbb{R}}^{\prime}}=\\{0\\}.$

For $\alpha\in{{}^{*}{\mathbb{Q}}}$ (non-standard rationals) such that
$\alpha\mathbf{u}\in{{}^{\prime}{\mathbb{R}}^{\prime}},$ define

$\mathrm{lm}_{\mathbb{U}}(\alpha\mathbf{u}):=-\pi\mathrm{st}(\alpha).$

This is an additive homomorphism

${{}^{\prime}{\mathbb{R}}^{\prime}}\twoheadrightarrow{\mathbb{R}}.$

For $\beta\in{{}^{*}{\mathbb{Q}}}$ such that
$\beta\mathbf{v}\in{{}^{\prime}i{\mathbb{R}}^{\prime}},$ define

$\mathrm{lm}_{\mathbb{U}}(\beta\mathbf{v}):=-i\pi\,\mathrm{st}(\beta),$

an additive homomorphism

${{}^{\prime}i{\mathbb{R}}^{\prime}}\twoheadrightarrow i{\mathbb{R}}.$

This defines $\mathrm{lm}_{\mathbb{U}}$ on
${{}^{\prime}{\mathbb{R}}^{\prime}}+{{}^{\prime}i{\mathbb{R}}^{\prime}}$
respecting the commutation with $\exp.$ Moreover,

$\mathrm{lm}_{\mathbb{U}}:\
{{}^{\prime}{\mathbb{R}}^{\prime}}+{{}^{\prime}i{\mathbb{R}}^{\prime}}\twoheadrightarrow{\mathbb{C}}.$

Since ${{}^{\prime}{\mathbb{R}}^{\prime}}$ and
${{}^{\prime}i{\mathbb{R}}^{\prime}}$ are divisible subgroups of
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}$ there is a complement subgroup
${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}(\infty),$

${\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}={{}^{\prime}{\mathbb{R}}^{\prime}}\dot{+}{{}^{\prime}i{\mathbb{R}}^{\prime}}\dot{+}{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}(\infty).$

Define $\mathrm{lm}_{\mathbb{U}}(u)=\infty$ for all
$u\in{\mathbb{U}}_{\mathfrak{p},\mathfrak{l}}(\infty).$ $\Box$

##  4 Integration

Recall that $\mathfrak{l}=\mu^{2}$ and $\mathfrak{i}=\iota^{2}$ elements of
${{}^{*}\mathbb{Z}}_{+}.$

###### 4.1

Proposition. Let $a=\frac{d^{2}}{l^{2}}$ for some $d,l\in{\mathbb{Z}}_{+}.$

$\sum_{-\mathfrak{i}\mathfrak{l}/2a\leq
n\leq\mathfrak{i}\mathfrak{l}/2a}\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{u})=\mu\frac{\iota\omega}{\sqrt{a}}$
(27)

$\sum_{-\mathfrak{l}/2a\leq
n\leq\mathfrak{l}/2a}\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{v})=\mu\frac{\omega}{\sqrt{a}}$
(28)

where $\omega\in\mathrm{F}_{\mathfrak{p}}$ is a primitive root of $1$ of order
$8.$ and $\mu,\iota$ and $d,l$ should be seen as elements of
$\mathrm{F}_{\mathfrak{p}}$ corresponding to the respective integers.

Proof. Let $\nu\in{{}^{*}\mathbb{Z}}_{+}$ be even and
$2\nu^{2}|(\mathfrak{p}-1).$ Let $\xi\in\mathrm{F}_{\mathfrak{p}}$ be a
primitive root of order $2\nu^{2},$ equivalently, for some
$\epsilon\in\mathrm{F}_{\mathfrak{p}}$, a primitive root of order
$\mathfrak{p}-1,$

$\xi=\epsilon^{\frac{\mathfrak{p}-1}{2\nu^{2}}}.$

Writing $n=m\nu+k,$ $0\leq m,k<\nu,$ we get

$\sum_{0\leq n<\nu^{2}}\xi^{n^{2}}=\sum_{0\leq
k<\nu}\xi^{k^{2}}\cdot\sum_{0\leq m<\nu}\xi^{m^{2}\nu^{2}+2mk\nu}.$ (29)

Now we use the fact that $\frac{m^{2}-m}{2}\in{{}^{*}\mathbb{Z}}$ and get in
$\mathrm{F}_{\mathfrak{p}}$

$\sum_{0\leq m<\nu}\xi^{m^{2}\nu^{2}+2mk\nu}=\sum_{0\leq
m<\nu}\xi^{m{\nu^{2}}+2mk\nu}=\sum_{0\leq m<\nu}\xi^{2m\nu(\frac{\nu}{2}+k)}=$

$=\\{\begin{array}[]{ll}{\nu},\mbox{ if }\frac{\nu}{2}+k\equiv 0\mod\nu,\\\
0,\mbox{ otherwise}\end{array}$

using that $\xi^{2m\nu(\frac{\nu}{2}+k)}=\xi^{2\nu^{2}}=1$ in the first line
and that

$\sum_{0\leq m<\nu}\zeta^{m}=0,$

for $\zeta:=\xi^{2\nu(\frac{\nu}{2}+k)},$ $\zeta^{\nu}=1$ but $\zeta\neq 1,$
in the second line.

Hence in the sum (29) only $k=\frac{\nu}{2}$ contributes, and we get

$\sum_{0\leq
n<\nu^{2}}\xi^{n^{2}}={\nu}\,\xi^{\frac{\nu^{2}}{4}}=\nu\epsilon^{\frac{\mathfrak{p}-1}{8}}=\nu\cdot\omega$
(30)

for $\omega:=\epsilon^{\frac{\mathfrak{p}-1}{8}},$ a primitive root of 1 of
order $8.$

Note that

$\sum_{0\leq n<\nu^{2}}\xi^{n^{2}}=\sum_{-\nu^{2}/2\leq
n<\nu^{2}/2}\xi^{n^{2}}$

because of periodicity

$\xi^{(n+\nu^{2})^{2}}=\xi^{n^{2}}.$

Finally, set

$\nu:=\frac{\mu\iota}{\sqrt{a}}\mbox{ for (\ref{sumu}) \ and\ }\
\nu:=\frac{\mu}{\sqrt{a}}\mbox{ for (\ref{sumv})}$

and (27) and (28) follow. $\Box$

###### 4.2

Corollary. For $\mathrm{lm}_{\mathrm{F}}$ of Theorem 3.13

$\mathrm{lm}_{\mathrm{F}}:\ \frac{1}{\mu}\sum_{-\mathfrak{i}\mathfrak{l}/a\leq
n<\mathfrak{i}\mathfrak{l}/a}\exp_{\mathfrak{p}}(a\frac{n^{2}}{2\mathfrak{l}}\mathbf{u})\
\ \mapsto\ \ \frac{1}{\sqrt{a}}$ $\mathrm{lm}_{\mathrm{F}}:\
\frac{1}{\mu}\sum_{-\mathfrak{l}/a\leq
n<\mathfrak{l}/a}\exp_{\mathfrak{p}}(a\frac{n^{2}}{2\mathfrak{l}}\mathbf{v})\
\ \mapsto\ \ \frac{e^{\frac{\pi i}{4}}}{\sqrt{a}}$

This follows from the facts that
$\mathrm{lm}_{\mathrm{F}}(\iota)=e^{-\frac{\pi i}{4}}$ and that
$\mathrm{lm}_{\mathrm{F}}(\epsilon^{\frac{\mathfrak{p}-1}{8}})=e^{\frac{\pi
i}{4}},$ see (25) and (26) together with Theorem 3.1.

###### 4.3

Note that $\frac{1}{\sqrt{a}}=\int_{\mathbb{R}}e^{-ax^{2}}dx,$ the classical
Gaussian integral. Analogously, in quantum mechanics
$\int_{\mathbb{R}}e^{iax^{2}}dx:=\frac{e^{\frac{\pi i}{4}}}{\sqrt{a}},$
although the integral is not classically defined since $e^{iax^{2}}$ is
oscillating on the whole of ${\mathbb{R}}.$ One of the ways of justifying the
assignment of the value to the integral expression is by referring to the fact
that the respective Fresnel integral $\int_{-A}^{A}e^{iax^{2}}dx$ is well-
defined for any $A>0$ and

$\lim_{A\to\infty}\int_{-A}^{A}e^{iax^{2}}dx=\frac{e^{\frac{\pi
i}{4}}}{\sqrt{a}}.$

###### 4.4

The domains of integrations and domains of summation.

Let, for $l\in{\mathbb{Z}}_{+}$

$\mathrm{I}_{l}=\\{n\in{{}^{*}\mathbb{Z}}:\ -l\mu\leq n\leq l\mu\\}\mbox{ and
}\mathrm{I}=\bigcup_{l\in{\mathbb{Z}}_{+}}\mathrm{I}_{l}.$

Let $a=\frac{m}{l}$ and

$\mathrm{I}_{a,\mathbf{u}}=\\{n\in{{}^{*}\mathbb{Z}}:-\mathfrak{i}\mathfrak{l}/2a\leq
n\leq\mathfrak{i}\mathfrak{l}/2a\\}\mbox{ and
}\mathrm{I}_{a,\mathbf{v}}=\\{n\in{{}^{*}\mathbb{Z}}:-\mathfrak{l}/2a\leq
n\leq\mathfrak{l}/2a\\}$

the domains of summations of (27) and (28). Clearly, using the assumptions on
$\mathfrak{l}$ and $\mu,$

$\mathrm{I}\subset\mathrm{I}_{a,\mathbf{u}}\mbox{ and
}\mathrm{I}\subset\mathrm{I}_{a,\mathbf{v}}$

and

$\frac{\mathbf{u}}{\mu}\cdot\mathrm{I}\subset{{}^{\prime}{\mathbb{R}}^{\prime}}\mbox{
and
}\frac{\mathbf{v}}{\mu}\cdot\mathrm{I}\subset{{}^{\prime}i{\mathbb{R}}^{\prime}}.$

The application of $\mathrm{lm}_{\mathbb{U}}$ defined in 3.13 gives us

$\mathrm{lm}_{\mathbb{U}}:\
\frac{\mathbf{u}}{\mu}\cdot\mathrm{I}_{l}\twoheadrightarrow{\mathbb{R}}\cap[-l\pi,l\pi]\mbox{
and }\frac{\mathbf{u}}{\mu}\cdot\mathrm{I}\twoheadrightarrow{\mathbb{R}},$
$\mathrm{lm}_{\mathbb{U}}:\
\frac{\mathbf{v}}{\mu}\cdot\mathrm{I}_{l}\twoheadrightarrow
i({\mathbb{R}}\cap[-l\pi,l\pi])\mbox{ and
}\frac{\mathbf{v}}{\mu}\cdot\mathrm{I}\twoheadrightarrow i{\mathbb{R}},$

that is $\frac{\mathbf{u}}{\mu}\cdot\mathrm{I}$ can be seen as a Riemann
integration partition of sets ${}^{\prime}{\mathbb{R}}^{\prime}$ with the
infinitesimal mesh $\frac{\mathbf{u}}{\mu},$ and respectively
$\frac{\mathbf{v}}{\mu}\cdot\mathrm{I}$ in $i{\mathbb{R}}$ with mesh
$\frac{\mathbf{v}}{\mu}.$

Note that in (27) and (28)

$\frac{n^{2}}{2\mathfrak{l}}=\frac{1}{2}(\frac{n}{\mu})^{2};\ \
\mathrm{lm}_{\mathbb{U}}(\frac{n^{2}}{2\mathfrak{l}}\mathbf{u})=-\pi\frac{x^{2}}{2},\
\
\mathrm{lm}_{\mathbb{U}}(\frac{n^{2}}{2\mathfrak{l}}\mathbf{v})=-i\pi\frac{x^{2}}{2}$

for $x=\mathrm{st}(\frac{n}{\mu}).$

###### 4.5

Lemma. For every $l\in{\mathbb{Z}}_{+}$

$\mathrm{lm}_{\mathrm{F}}:\frac{1}{\mu}\sum_{n\in\mathrm{I}_{l}}\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{u})\mapsto\int_{-l}^{l}e^{-ax^{2}}dx$
$\mathrm{lm}_{\mathrm{F}}:\frac{1}{\mu}\sum_{n\in\mathrm{I}_{l}}\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{v})\mapsto\int_{-l}^{l}e^{-iax^{2}}dx$

and the integrals are well-defined.

Proof. Note that by (19) and (20) the map $\mathcal{I}$ translates the
respective elements of the sums from $\mathrm{F}_{\mathfrak{p}}$ to the
elements of the non-standard model ${{}^{*}{\mathbb{C}}}$ of complex numbers:

$\mathcal{I}:\
\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{u})\mapsto
e^{-a\pi(\frac{n}{\mu})^{2}}$ $\mathcal{I}:\
\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{v})\mapsto
e^{-ia\pi(\frac{n}{\mu})^{2}}$

and thus

$\mathcal{I}\\{\frac{1}{\mu}\sum_{n\in\mathrm{I}_{l}}\exp_{\mathfrak{p}}(-a\pi\frac{n^{2}}{2\mathfrak{l}}\mathbf{u})\\}\mbox{
and
}\mathcal{I}\\{\frac{1}{\mu}\sum_{n\in\mathrm{I}_{l}}\exp_{\mathfrak{p}}(-a\pi\frac{n^{2}}{2\mathfrak{l}}\mathbf{v})\\}$

become non-standard Riemann sums with infinitesimal mesh $\frac{1}{\mu}$

$\sum_{-l<\frac{n}{\mu}<l}\frac{1}{\mu}e^{-a\pi(\frac{n}{\mu})^{2}}\mbox{ and
}\sum_{-l<\frac{n}{\mu}<l}\frac{1}{\mu}e^{-ia\pi(\frac{n}{\mu})^{2}}$

Since the summation is over a compact interval
$[-l,l]\subset{{}^{*}{\mathbb{R}}}$ the application of the standard part map
gives us (see e.g. the integration via non-standard analysis in [12])

$\mathrm{st}(\sum_{-l<\frac{n}{\mu}<l}\frac{1}{\mu}e^{-a\pi(\frac{n}{\mu})^{2}})=\int_{-l}^{l}e^{-a\pi
x^{2}}dx\mbox{ and
}\mathrm{st}(\sum_{-l<\frac{n}{\mu}<l}\frac{1}{\mu}e^{-ia\pi(\frac{n}{\mu})^{2}})=\int_{-l}^{l}e^{-ia\pi
x^{2}}dx$

###### 4.6

Corollary.

$\mathrm{lm}_{\mathrm{F}}:\frac{1}{\mu}\sum_{n\in\mathrm{I}}\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{u})\mapsto\int_{\mathbb{R}}e^{-ax^{2}}dx=\frac{1}{\sqrt{a}}$
$\mathrm{lm}_{\mathrm{F}}:\frac{1}{\mu}\sum_{n\in\mathrm{I}}\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{v})\mapsto\lim_{l\to\infty}\int_{-l}^{l}e^{-iax^{2}}dx=\frac{e^{\frac{\pi}{4}}}{\sqrt{a}}$

Proof. In both cases the right-hand side is the limit of integrals in 4.5
since $\mathrm{I}=\bigcup_{l}\mathrm{I}_{l}.$ In the first case the classical
Gaussian integral over the whole of ${\mathbb{R}}$ converges and is equal to
the limit. In the second case the right-hand side as the limit of the Fresnel
integral is well-defined but the Riemann integral is not.

###### 4.7

Discussion. The left-hand sides of 4.2 and of 4.6 differ in the domains of
summations but the right-had sides are the same. This implies that our
definition of $\mathrm{lm}_{\mathrm{F}}$ is such that

$\mathrm{lm}_{\mathrm{F}}\left(\frac{1}{\mu}\sum_{n\in\mathrm{I}_{a,\mathbf{u}}\setminus\mathrm{I}}\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{u})\right)=0\mbox{
and
}\mathrm{lm}_{\mathrm{F}}\left(\frac{1}{\mu}\sum_{n\in\mathrm{I}_{a,\mathbf{v}}\setminus\mathrm{I}}\exp_{\mathfrak{p}}(-a\frac{n^{2}}{2\mathfrak{l}}\mathbf{v})\right)=0$

In these “tail domains” $\mathrm{I}_{a,\mathbf{u}}\setminus\mathrm{I}$ and
$\mathrm{I}_{a,\mathbf{v}}\setminus\mathrm{I}$ the respective values under
exponentiation are very large, non-feasible numbers, and according to the
inerpretation of $\mathrm{F}_{\mathfrak{p}}$ in ${\mathbb{C}}$ the application
of $\exp$ to such values oscillates uncontrollably. R.Feynman intuition was
that for this reason the sum should be considered negligible, see e.g. [13],
2-3.

## References

  * [1] B.Zilber, Perfect infinities and finite approximation, In: Infinity and Truth. IMS Lecture Notes Series, V.25, 2014
  * [2] J.A.Cruz Morales and B.Zilber, The geometric semantics of algebraic quantum mechanics, Phil. Trans. R. Soc. A 2015 373 20140245; DOI: 10.1098/rsta.2014.0245. Published 29 June 2015
  * [3] B.Zilber, The semantics of the canonical commutation relations arxiv1604.07745 (see also a 2016 version on author’s webpage)
  * [4] B.Zilber, Structural approximation and quantum mechanics, 2017, author’s web-page
  * [5] C.Yang and T.Lee, Statistical Theory of Equations of State and Phase Transition I. Theory of condensation, Phys.Rev. v87, 3, p.404, 1952
  * [6] C.Yang and T.Lee, Statistical Theory of Equations of State and Phase Transition II. Lattice Gas and Ising Model, Phys.Rev. v87, 3, p.410, 1952
  * [7] D.Tong, Statistical Physics, Preprint, Cambridge, https://www.damtp.cam.ac.uk/user/tong/statphys.html
  * [8] V.Sazonov, On feasible numbers, International Workshop on Logic and Computational Complexity, LCC 1994: Logic and Computational Complexity, pp 30 - 51
  * [9] E.Zeidler, Quantum Field Theory v.1 ( P.3 The Challenge of Different Scales in Nature)
  * [10] P. Kustaanheimo, On the fundamental prime of the finite world,Annales Academiae scientiarum Fennicae. Series A. 1, Mathematica-Physica, 129, 1952
  * [11] E. Welti, Die Philosophie des strikten Finitismus. Entwicklungstheoretische und mathematische Untersuchungen über Unendlichkeitsbegriffe in Ideengeschichte und heutiger Mathematik, Bern: Peter Lang, 1987.
  * [12] A. Robinson, Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966.
  * [13] R.Feynman and A.Hibbs, Quantum Machanics and Path Integrals, Dover Publ., 2005
  * [14] J.-H.Evertse The number of solutions of linear equations in roots of unity, Acta Arithmetica (1999) v89, 1, pp. 45 - 51