The extent of the central structure can be understood from a comparison with the propagation of the wave packages in the top panel of <ref>. As the wave-packages recede from their initial position in <ref>, no non-trivial dynamics occurs in the center of the simulation domain. Thus a coarser time spacing suffices in that region. Similarly, later, when the wave packages recombine to interfere, a field configuration very similarly to the initial conditions (except for a flipped sign) occurs and the temporal spacing reduces to resolve these non-trivial dynamics. With the wave-packages thereafter leaving the central region again to travel to the opposite end of the spatial domain there is also no more need for a high resolution and the ridge structure reappears.

The structures close to the boundary also correlate with the propagation of the wave-packages. I.e. the deviations from the constant values occurs exactly where the wave-packages reflect from the spatial boundary. The temporal spacing is not just reduced but shows an intricate mix of refinement and coarsening. As the wave-package hits the wall, the adaption of the temporal grid emerges fully dynamically.

Continuing the investigation of the time mapping, we inspect its spatial derivative in the bottom panel of <ref> next. As expected from our discussion above, the central region and the boundaries show non-trivial modifications. Non-trivial gradients emerge in the central region, which appear accumulative. They build up as the wave-packages recede from their initial positions, while their growth is stopped over the short interval where the wave-packages recombine to interfere. Similarly, after the wave-packages have hit the spatial boundary, the time-mapping develops gradients, which persist at larger $\tau$ and we expect these effects to accumulate with each encounter of the boundary by the wave-packages.

In summary, the results clearly show that in our approach with dynamical coordinate maps the temporal grid adapts automatically to the dynamics of the propagating fields, constituting a genuine form of automatic adaptive mesh refinement.

Let us continue with the second central result of our numerical study, the evaluation of the energy, i.e. the Noether charge associated with time-translation symmetry in our approach. In the top panel of <ref> we show the values of ${\bm Q}^{\rm L}_t$, defined in <ref>, which is associated with time-translations, a manifest symmetry of our discretized action <ref>. Its values are given as red circles. A visual inspection reveals that ${\bm Q}^{\rm L}_t$ remains constant throughout the whole simulation with no deviation from its initial value. Since ${\bm Q}^{\rm L}_t$ makes reference to first derivatives only, its value at initial $\tau^{\rm i}$ is determined solely by the initial conditions provided and that value persists throughout the simulation[While we can fix the value and $\tau$ derivatives of the degrees of freedom to their continuum values at $\tau^{\rm i}$, the values of the spatial derivatives follow self-consistently from an evaluation of $\mathds{D}_\sigma$ applied to the discretized initial values. I.e. when the initial energy is evaluated at different lattice spacings, the spatial derivative will encounter the same initial shape of e.g. the field bump at different resolutions, providing slightly different values for the energy there.].

(top) Preservation of the Noether charge ${\bm Q}^{\rm L}_t$ (red open circles) over the whole simulation domain. (bottom) The two individual contributions to the Noether charge from the coordinates (blue crosses) and from the field dynamics (red circles). Note that we have subtracted the initial value $\dot t_0$ from the coordinate contribution for better visibility of the mutually complementary behavior.

Values of the Noether charge ${\bm Q}^{\rm L}_t$ for different temporal extent of the simulation, implemented via different values of the initial $\tau$ derivative $\td$ of the time-mapping. Note that magnitude of the Noether charge depends on the initial value of $\dot t$.

The total energy of our system is composed of a contribution from the field, and in addition from the time mapping, according to <ref>. In the bottom panel of <ref> we plot the two contributions separately. To highlight the mutually complementary behavior we have subtracted the initial value $\dot t_{\rm IC}$ from the coordinate contribution. One finds that while the overall energy ${\bf Q}_t^{\rm L}$ is identically preserved, the coordinate and field contribution each show minute variation in time, which exactly cancel in their sum. This dependence of the energy on the coordinate mapping was expected, as it occurred in a similar fashion in the world-line formalism.

With its explicit coordinate mapping dependence, we showcase the values of ${\bm Q}^{\rm L}_t$ for different $\dot t_{\rm IC}$ in <ref>. The underlying resolution of the $\tau$ direction is kept constant and we increase $N_\tau$ such that approximately the same $t$ resolution is obtained. Due to our choice of a large value of $T=10^4$, the main contribution to the energy indeed comes from the time mapping itself, which is why the values of ${\bm Q}^{\rm L}_t$ increase with increasing $\td_{\rm IC}$. Note that the values are slightly larger than $\td$ and it is this difference that encodes the contributions from the field dynamics. We have explicitly checked that for all initial values $\dot t_{\rm IC}$ shown in <ref> the Noether charge remains at its initial value within the global numerical accuracy of our solution $\Delta \sim 10^{-6}$.

Having established that the Noether charges are exactly preserved, we can now propose an explanation as to the guiding principle underlying the automatic adaptive mesh refinement observed in our simulations. We argue that it is based on the preservation of the space-time symmetries of our approach and the conservation of the associated Noether charge. Only if the space-time grid adapts to the dynamics of the field, is it possible for a non-trivial Noether charge, such as ${\bm Q}_t^{\rm L}$ to be exactly preserved.

Note that while exact conservation of continuum Noether charges is a central feature of our novel approach, it is not synonymous with obtaining the continuum solution. For a finite grid spacing the solution will be affected by numerical error, which, as we demonstrate below, diminishes under grid refinement. The numerical errors however, by construction of our method, cannot affect the value of the Noether charges.

Let us briefly return to the fact that the explicit inclusion of initial, boundary and connecting conditions in our discretized action <ref> introduces non-trivial contributions to the values of the field and coordinate mappings at the spatial and temporal boundaries.

We find e.g. that the $\tau$ derivative of the time mapping $\dot t$ in the top panel of <ref> shows a jump at the last $\tau$ slice. This behavior is connected to the influence of the Lagrange multipliers there. The jump corresponds to a finite but small correction to the discretized equation of motion derived in the absence of Lagrange multipliers. Just as for the Noether charge it should be possible to define an expression for the time mapping in which the effect of the Lagrange multiplier is accounted for and for which the jump in the last time slice is absent.
We did not pursue the derivation of this correction term of $t$ further at this point. However, as we will show next, that the effects of the Lagrange multipliers do not affect the convergence of the solution under grid refinement.

The global $L_2$-norm difference of the solutions for $t$ (top) and field $\phi$ (bottom) obtained via the direct determination of the critical point of $\mathds{E}^{\rm L}_{\rm IBVP}$ and by solving <ref> on high resolution grids using the  method of 's .

To demonstrate that we obtain the correct convergence rate under grid refinement, we compare the time mapping $t$ and field $\phi$ obtained from our approach, i.e. from finding the critical point of $\mathds{E}_{\rm IBVP}^{\rm L}$ to solutions of the continuum equations of motion <ref> obtained via the  method of 's  on high resolution grids, corresponding to the default  of eight significant digits. The global $L_2$-norm differences obtained via integration over both $\tau$ and $\sigma$

\begin{align}
&\epsilon^t = \sqrt{ ({\bm t}_1-{\bm t}_{\rm NDSolve})^T \mathds{H}\, ({\bm t}_1-{\bm t}_{\rm NDSolve})},\\
&\epsilon^\phi = \sqrt{ ({\bm \phi}_1 - {\bm \phi}_{\rm NDSolve}) ^T \mathds{H} \,({\bm \phi}_1 - {\bm \phi}_{\rm NDSolve}) },
\end{align}

are shown in <ref>. The top panel shows $\epsilon^t$, the bottom panel $\epsilon^\phi$, each as red symbols. We concurrently reduce the spatial and temporal grid spacing and find that the deviation systematically reduces. The solid red line denotes the best power-law fit $\alpha \sqrt{\Delta \sigma^2+\Delta \tau^2}^\beta$ to the results. The fits tell us that for $\epsilon^t$ we have $\beta=1.89$ and for $\epsilon^\phi$ the scaling goes as $\beta=2.07$. Both of these values lie close to the ideal $\beta=2$, which corresponds to the ideal scaling of a  operator. We note that due to the use of the pre-compiled  library and the associated limitation to global numerical accuracy to around $\Delta\sim10^{-6}$ we did not grid refine further.

(top) The field $\phi(t,x)$ as function of the physical space-time coordinates $x$ and $t$. (bottom) The global $L_2$-norm difference between the solution for $\phi(x,t)$ obtained via the direct determination of the critical point of $\mathds{E}^{\rm L}_{\rm IBVP}$ and by solving the conventional wave equation on high resolution grids using the  method of 's .

The conventional energy $E_{\rm conv}$ defined in <ref>, as a function of time $t$ on solutions of the field that propagate to different final times $t^{\rm f}$. Note that $E_{\rm conv}$ is insensitive to the initial $\tau$ derivative of the time mapping $\dot t$, which we use to select the final time of our simulation.

For completeness let us also visualize the field solution in physical space-time coordinates, as shown in the top panel of <ref>. While the spatial domain is fixed at all times to have the same extent, due to the use of our trivial mapping $x=\sigma$, the temporal simulation domain extends to slightly different values depending on the location along $x$. Due to T being large, these effects here are relatively small, as we saw that the deviation from a constant value of the derivatives of $t(\tau,\sigma)$ were at most on the level of a percent.

We may ask how well the physical field configuration $\phi(x,t)$ obtained from our new approach agrees with the solution of the conventional wave equation (WE) in (<ref>). To this end we solve the wave equation via the  method of  in  on high resolution grids, to achieve the  of eight significant digits. This numerical solution is denoted as ${\bm \phi}_{\rm WE}$.

Due to the presence of the square root, the novel action contains additional terms, which are however suppressed by powers of $T$. For our choice of $T=10^4$ we find that for all the grid spacings considered in this study, convergence to the numerical wave equation solution ${\bf \phi}_{\rm WE}$ in the global $L_2$ norm

\begin{align}
&\epsilon^\phi_{\rm WE} = \sqrt{ ({\bm \phi}_1 - {\bm \phi}_{\rm WE}) ^T \mathds{H} \,({\bm \phi}_1 - {\bm \phi}_{\rm WE}) }
\end{align}

can be found under grid refinement, as shown in the bottom panel of <ref>. Of course, as one increases the grid resolution further one would eventually spot differences between our solution and that of the wave equation when the numerical error becomes smaller than the first correction term involving $T$.

It is also instructive to evaluate the conventional energy in the field $\phi$

\begin{align}
\nonumber E_{\rm conv}&=\int dx \frac{1}{2}\Big\{ \Big( \frac{\partial \phi}{\partial t}\Big)^2 + \Big( \frac{\partial \phi}{\partial x}\Big)^2 \Big\}\\
&\approx \frac{1}{2}\mathds{H}_\sigma \Big\{ \Big( {\mathds{D}_\tau}{\bm \phi}/  {\mathds{D}_\tau}{\bm t} \Big)^2 + \Big( {\mathds{D}_\sigma}{\bm \phi}/  {\mathds{D}_\sigma}{\bm x} \Big)^2\Big\}\label{eq:convnrg}
\end{align}

where division is understood in a point-wise fashion. The values of <ref> for different physical time extent are shown in <ref>. Note first, that in contrast to ${\bf Q}_t^{\rm L}$ this quantity at $t=0$ does not vary with $\dot t_{\rm IC}$, since the conventional energy only knows of the field itself. In addition we see that while the values of the conventional energy seem to stay close to a common value on average, the individual values of $E_{\rm conv}$ show significant oscillatory patterns (and a jump from the value at $t=0$). It is clearly not the actual conserved Noether charge associated with time translation symmetry.

While a thorough investigation of the scaling properties of our approach for  operators is planned for a future study, let us briefly confirm that the conservation of the Noether charge indeed holds even for the higher order SBP discretization scheme. I.e. the exact conservation observed in <ref> is not an accident of the lowest order with its very simple boundary modifications within $\mathds{D}$. Hence, in <ref> we show the values of ${\bm Q}^{\rm L}_t$ on a $N_\sigma=24$ and $N_\tau=16$ grid obtained from the  scheme as red circles.  Note that also for , as long as the Lagrange multiplier contributions to the Noether charge are correctly incorporated according to <ref>, we obtain exactly conserved values of the Noether charge.

The values of the Langrange multiplier corrected ${\bm Q}^{\rm L}_t$ Noether charge from the  operator on a grid with $N_\tau=16$ and $N_\sigma=24$. Also for the higher order operator we confirm that within our accuracy the Noether charge is exactly preserved.

§ SUMMARY AND OUTLOOK

We have presented two novel conceptual contributions to address central challenges of formulating IBVPs.

§.§ Summary

The first novel contribution is the derivation of a reparameterization invariant continuum action, in which coordinate maps are incorporated as dynamical degrees of freedom. A new energy density scale $T$ determines when the dynamics of the field become inextricably intertwined with that of the coordinate maps.

In the novel action we do not integrate over space-time coordinates but instead over a set of underlying abstract parameters $\Sigma^a$. This fact leads to the second novel contribution, the ability to discretize the action in terms of the abstract parameters, leaving the values of the coordinate maps continuous. In turn the discretized action retains manifest invariance under infinitesimal Poincaré transformations and Noether's theorem remains intact. Using summation-by-parts (SBP) operators, the Noether charges are exactly preserved in time.

To formulate a causal initial boundary value problem we deployed the doubled degrees-of-freedom construction of Galley, to obtain the IBVP action. The requires initial, boundary and connecting conditions are made explicit by introducing appropriate sets of Lagrange multiplier functions in the continuum.

Deploying appropriately regularized summation-by-parts finite difference operators formulated in affine coordinates, we outlined how to discretize the novel system action and arrive at the correct expression for the discrete Noether charge.

<Ref> presented a proof-of-principle based on wave propagation in $1+1$ dimensions deploying a fully dynamic time mapping $t(\tau,\sigma)$ and a propagating field $\phi(\tau,\sigma)$. We showed how our approach offers new flexibility in boundary treatment.

Our main numerical results are as follows:

* The coordinate map $t$ adapts dynamically to the evolution of the field $\phi$, exhibiting region of coarser and finer resolution, correlated with the field dynamics. This constitutes a form of automatic adaptive mesh refinement.

* The Noether charge of time translation symmetry, is exactly conserved at its initial value. This energy receives contributions from the field and the coordinate map. We find that the conventional energy density on the other hand shows oscillatory patterns, as it is not the correct conserved charge.

* The novel approach implemented with  operators converges correctly under grid refinement.

* We have confirmed that the Noether charge is exactly conserved also for a higher order  scheme.

§.§ Future work

Our proof-of-principle explored the dynamics of the time-mapping $t(\tau,\sigma)$ but kept the spatial mapping trivial as $x=\sigma$. The inclusion of a fully dynamic spatial map $x(\tau,\sigma)$ is work in progress. While formally straight forward, one has to take into account that for fully dynamic coordinate maps the action contains additional redundancies. In the string-theory literature these are addressed by selecting e.g. conformal gauge.

As more flexible coordinate maps are introduced also the freedom in the boundary terms increases with more degrees of freedom contributing to their values. We are working on exploring whether there exist compatibility requirements for the boundary conditions, especially in the presence of a specific gauge choice.

Our proof of principle was implemented in $1+1$ dimensions and an extension to higher dimensions is called for. The numerical minimisation underlying the search for the critical point $\mathds{E}^{\rm L}_{\rm IBVP}$ will become more costly. This will eventually demand a departure from the  software and implementation in a more performant language, such as e.g.  via the  library, which is work in progress.

In this study we have enforced initial, boundary, and connecting conditions in a strong manner with Lagrange multipliers. Weak imposition of boundary data offers many advantages, especially for numerical stability. It is therefore of interest to formulate a genuine weak version of our action based approach.

We believe that the preservation of the Noether charge is closely related to stability. In future work we will elaborate on this and will attempt a proof.

As explained but not elaborated on, the presence of $\tau$ and $\sigma$ derivatives on the coordinate fields opens up for new constructions of non-reflecting boundary conditions. We will explore that possibility for the wave equation in future work.

§.§ Speculations

It is natural to ask, whether a similar reparameterization invariant action as in the scalar case, can be constructed for other relevant theories, such as gauge fields (linear Maxwell electromagnetism, non-linear Yang-Mills theory). Treatment of gauge fields in string theory via the Born-Infeld construction suggests that such a construction is not unique but may be equivalent to a low order in the non-relativistic expansion.

So far the benefits of our novel action have been elucidated in the context of classical field theory. The preservation of space-time symmetries plays a central, if not more crucial role, for the accurate simulation of quantum fields. As breaking of space-time symmetries directly translates into a contamination of the particle spectrum observed in these simulations with unphysical modes, exact space-time symmetry preservation promises significant improvements in that context.

The novel action we propose motivates several lines of inquiry on the physics side. One may ask what the role of the new scale $T$ is, which was central to the construction of $\cal S$. In the world-line formalism, its counterpart $mc$ is associated with the rest energy of a point particle, a fundamental property of the propagating degree of freedom. Here $T$ seems to lend itself to the interpretation of a reference energy density for the coordinate dynamics and it is intriguing to investigate its possible relation to the physics of gravity and the cosmological constant through the general theory of relativity and string theory.

We may further ask about possible physics implications of the square root term in the fully reparameterization invariant action with dynamical coordinate maps. If $\cal S$ indeed were the correct high-energy action for second order field theory, then at low energies, small but finite correction terms to the conventional $S_{\rm ft}$ will ensue. This hypothesis should be straight forward to falsify, by exploring whether such corrections are compatible with the wealth of high precision collider based measurements available for the fundamental interactions on short scales and via astronomical observations on large scales.

§ ACKNOWLEDGEMENTS

A. R. and W. A. H. acknowledge support by the ERASMUS+ project 2023-1-NO01-KA171-HED-000132068. W. A. H. thanks the South African National Research Foundation and the SA-CERN Collaboration for financial support. J. N. was supported by the Swedish Research Council grant nr. 2021-05484 and the University of Johannesburg.

§ DERIVATION OF THE CONNECTING CONDITIONS

In the following we will derive how the action <ref> and connecting conditions of <ref> prevent the appearance of non-causal temporal boundary terms. To this end it is convenient to express ${\cal E}_{\rm IBVP}$ in linear combinations of $X_1,X_2$ and $\phi_1,\phi_2$. We consider the difference $X_-=X_1-X_2$, $\phi_-=\phi_1-\phi_2$ and the mean $X_+=\frac{1}{2}(X_1+X_2)$, $\phi_+=\frac{1}{2}(\phi_1+\phi_2)$ (known in the context of the Schwinger-Keldysh literature also as the quantum and classical components respectively). Let us inspect the variation of the action w.r.t. these new variables

\begin{align}
\delta {\cal E}_{\rm IBVP}[X_\pm,& \partial_a X_\pm, \phi_\pm, \partial_a \phi_\pm]\\
\nonumber=\int d^{(d+1)}\Sigma \Big\{
&\frac{\partial E_{\rm IBVP}}{\partial X_+^\mu}\delta X_+^\mu + \frac{\partial E_{\rm IBVP}}{\partial (\partial_a X_+^\mu)}\delta (\partial_a X_+^\mu) + \frac{\partial E_{\rm IBVP}}{\partial \phi_+}\delta \phi_+ + \frac{\partial E_{\rm IBVP}}{\partial (\partial_a \phi_+)}\delta (\partial_a \phi_+) \\
+& \frac{\partial E_{\rm IBVP}}{\partial X_-^\mu}\delta X_-^\mu + \frac{\partial E_{\rm IBVP}}{\partial (\partial_a X_-^\mu)}\delta (\partial_a X_-^\mu) + \frac{\partial E_{\rm IBVP}}{\partial \phi_-}\delta \phi_- + \frac{\partial E_{\rm IBVP}}{\partial (\partial_a \phi_-)}\delta (\partial_a \phi_-)\Big\}\label{eq:varEIVP}\\
\nonumber=\int d^{(d+1)}\Sigma \Big\{&
\Big( \frac{\partial E_{\rm IBVP}}{\partial X_+^\mu} -\partial_a \frac{\partial E_{\rm IBVP}}{\partial (\partial_a X_+^\mu)}\Big)\delta X_+^\mu + \Big(  \frac{\partial E_{\rm IBVP}}{\partial \phi_+} - \partial_a \frac{\partial E_{\rm IBVP}}{\partial (\partial_a \phi_+)} \Big)\delta \phi_+ \\
\Big( \frac{\partial E_{\rm IBVP}}{\partial X_-^\mu} -\partial_a \frac{\partial E_{\rm IBVP}}{\partial (\partial_a X_-^\mu)}\Big)\delta X_-^\mu + \Big(  \frac{\partial E_{\rm IBVP}}{\partial \phi_-} - \partial_a \frac{\partial E_{\rm IBVP}}{\partial (\partial_a \phi_-)} \Big)\delta \phi_-\Big\} \label{eq:varEeomIVP}\\
+\int &\prod_{a=1}^{d}d\Sigma_a\Big\{ \Big[ \frac{\partial E_{\rm IBVP}}{\partial (\partial_0 X_+^\mu)} \delta X_+^\mu \Big]_{\tau^{\rm i}}^{\tau^{\rm f}} + \Big[ \frac{\partial E_{\rm IBVP}}{\partial (\partial_0 X_-^\mu)} \delta X_-^\mu \Big]_{\tau^{\rm i}}^{\tau^{\rm f}} \label{eq:tauBCEvarXIVP} \\
&\qquad \quad   + \Big[ \frac{\partial E_{\rm IBVP}}{\partial (\partial_0 \phi_+)} \delta \phi_+ \Big]_{\tau^{\rm i}}^{\tau^{\rm f}} + \Big[ \frac{\partial E_{\rm IBVP}}{\partial (\partial_0 \phi_+)} \delta \phi_+ \Big]_{\tau^{\rm i}}^{\tau^{\rm f}} \Big\}\label{eq:tauBCEvarphiIVP}\\
\nonumber + {\rm spatial\, B.C.}
\end{align}

For the discussion of causality, the spatial boundary conditions do not play a role, which is why we do not list them here explicitly. Prescribing fixed initial conditions for the values of the d.o.f. in our system, we have $\delta X^\mu_\pm(\tau^{\rm i},\vec{\sigma})=0$ and  $\delta \phi_\pm(\tau^{\rm i},\vec{\sigma})=0$, making the lower boundaries of the terms in (<ref>) and (<ref>) vanish.

At the end of the forward branch at $\tau^{\rm f}$ we can achieve the vanishing of some of the boundary terms by requiring that the values of the degrees of freedom on the forward and backward branch agree

\begin{align}
X_1^\mu(\tau=\tau^{\rm f},\vec{\sigma})=X_2^\mu(\tau=\tau^{\rm f},\vec{\sigma}),\quad \phi_1(\tau=\tau^{\rm f},\vec{\sigma})=\phi_2(\tau=\tau^{\rm f},\vec{\sigma}).\label{eq:connval}
\end{align}

This leads to $\delta X^\mu_-(\tau^{\rm f},\vec{\sigma})=0$ and  $\delta \phi_-(\tau^{\rm f},\vec{\sigma})=0$. The only remaining terms are those involving $X_+^\mu$ and $\phi_+$ at the final $\tau^{\rm f}$.

Since for a causal IBVP the values of the d.o.f. are not a priori known at the final time $\tau^{\rm f}$, the corresponding variation in (<ref>) and (<ref>) cannot be made to vanish there. Let us instead inspect the terms which multiply the variations, which in this case are $\partial E_{\rm IBVP}/\partial (\partial_0 X_+^\mu)$ and $\partial E_{\rm IBVP}/\partial (\partial_0 \phi)$. Since the variations for each $\mu$ component of $X$ are independent, we can consider just one of them, e.g. $\mu=d$, to establish how the boundary contribution for each and all of them can be made to vanish.
By exploiting the inverse relations $X^\mu_1=X^\mu_+ + \frac{1}{2} X^\mu_-$ and $X^\mu_2=X^\mu_+ - \frac{1}{2} X^\mu_-$ we find that

\begin{align}
\nonumber\frac{\partial E_{\rm IBVP}}{\partial (\partial_0 X_+^d)} &=  \frac{\partial E_{\rm IBVP}}{\partial (\partial_0 X_1^d)} \frac{\partial_0 X_1^d}{\partial_0 X_+^d}+\frac{\partial E_{\rm IBVP}}{\partial (\partial_0 X_2^d)} \frac{\partial_0 X_2^d}{\partial_0 X_+^d}\\
&=  \frac{\partial E_{\rm BVP}[X_1,\partial_a X_1,\phi_1,\partial_a\phi_1]}{\partial (\partial_0 X_1^d)} -\frac{\partial E_{\rm BVP}[X_2,\partial_a X_2,\phi_2,\partial_a\phi_2]}{\partial (\partial_0 X_2^d)}\label{eq:tbnd0}
\end{align}

As one can confirm in <ref>, $E_{\rm BVP}$ makes reference to $X$ in the terms involving ${\rm det}[g]$ and ${\rm adj}[g]_{ab}$.

It is important to realize that in <ref> each of the two terms refers to a different induced metric for the forward and backward branch respectively. Thus the full adjugate of the induced metric needs to be the same between the two branches for cancellation to occur. This in turn requires that at final $\tau^{\rm f}$ all the different derivatives of $X^\mu$ must agree on the two branches.
We thus conclude that in order for the boundary terms related to the $X_+$ variables to vanish, we must require

\begin{align}
\partial_a X_1^\mu|_{\tau=\tau^{\rm f}}= \partial_a X_2^\mu|_{\tau=\tau^{\rm f}}.\label{eq:coordSKderiv}
\end{align}

For the field degrees of freedom we have to consider instead

\begin{align}
\nonumber\frac{\partial E_{\rm IBVP}}{\partial (\partial_0 \phi_+)} &=  \frac{\partial E_{\rm IBVP}}{\partial (\partial_0 \phi_1)} \frac{\partial_0 \phi_1}{\partial_0 \phi_+}+\frac{\partial E_{\rm IBVP}}{\partial (\partial_0 \phi_2)} \frac{\partial_0 \phi_2}{\partial_0 \phi_+}\\
&=  \frac{\partial E_{\rm BVP}[X_1,\partial_a X_1,\phi_1,\partial_a\phi_1]}{\partial (\partial_0 \phi_1)} -\frac{\partial E_{\rm BVP}[X_2,\partial_a X_2,\phi_2,\partial_a\phi_2]}{\partial (\partial_0 \phi_2)}.
\end{align}

In turn from the field dependent terms in <ref> one is led to expressions such as

\begin{align}
\frac{\partial}{\partial (\partial_0 \phi_1)} \partial_a\phi_1\partial_b\phi_1 {\rm adj}[g]_{ab}= 2\, {\rm adj}[g]_{0b}\partial_b\phi_1
\end{align}

Since we have made sure by (<ref>) that the adjugate of the induced metric agrees betweeen the forward and backward branch at final $\tau$, we can thus write

\begin{align}
\frac{\partial E_{\rm BVP}}{\partial (\partial_0 \phi_1)} -\frac{\partial E_{\rm BVP}}{\partial (\partial_0 \phi_2)}= 2 {\rm adj}[g]_{0b}\Big( \partial_b\phi_1 - \partial_b\phi_2\Big).
\end{align}

The adjugate of the induced metric is in general a dense matrix. In order for the above expression to vanish, we find that it is necessary to identify the derivatives of the field with respect to all $\Sigma^a$ at final $\tau$

\begin{align}
\partial_a \phi_1|_{\tau=\tau^{\rm f}}= \partial_a \phi_2|_{\tau=\tau^{\rm f}}.\label{eq:fieldSKderiv}
\end{align}

The outcome of the above derivation entails that we need to fix all $(d+1)$ derivatives where the forward and backwards path meet. However in the continuum formalism, the fact, that we identify the values of our degrees of freedom at $\tau^{\rm f}$ already implies that all spatial derivatives too are the same[In the discrete setting, depending on the choice of stencil used for the derivative approximation, it may become necessary to introduce individual constraints for the spatial derivatives]. Then it is only the derivatives in $\tau$ that we need to enforce to be equal.

The combination of <ref> constitutes the connecting conditions summarized in the main text in <ref>.

§ COMPETING INTERESTS

The authors declare that they have no competing interests.

§ AUTHOR'S CONTRIBUTIONS

* A. Rothkopf: construction of the novel action (contribution based on the world-line formalism and field theory), numerical implementation, writing, editing

* W. A. Horowitz: construction of the novel action (contribution based on string theory and field theory), editing

* J. Nordström: cross-validating of the novel approach (contribution based on applied analysis), editing

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