A coherent state is one with $\mathrm{\Delta }X\mathrm{\Delta }P=\text{minimum}$ and maximal entropy while the first and second moments of position and momentum are fixed. For an excellent review of this subject, see [16].

Note also that this technique can be employed to build up an analytical perturbation series in ${\mathit{H}}_R$; as we will be interested in strongly coupled dynamics, I do not develop this direction here.

The energy is a function of $X_0$ and does not depend on $P_0$. In Matrix theories with supersymmetry, the contribution from the fermions typically adds momentum dependence as well. We would correspondingly need an expanded version of this equation.

This idea was inspired by previous computations [27, 28]. I also note some similarities of our proposal with some of the suggestions in the work of [29].

It is interesting to see the role of the measuring process in the emergence phenomenon. For example, an infinitely accurate position measurement, $\xi \to 0$, would lead to the zero temperature limit at constant entropy. Looking at the quadratic part of the effective Hamiltonian (35), the eigenfrequencies can be found to be ${\omega }_1=g\sqrt{\xi (d-1)}/2$ for two modes ($d$ being the dimension of the target space), and $d-2$ modes with ${\omega }_2=\sqrt{g^2\xi (d-1)/2+g^2{X}_0^2}\simeq g{X}_0$. ${\omega }_1\to 0$ is typically identified with the zero modes arising from commuting matrices, a well-known instability of BFSS Matrix theory [18, 33, 34]. We see that, using a more realistic measurement paradigm that introduces coherent states also naturally regularizes this instability.