Notice the Thomas-Yau argument is a little mysterious from the following classical perspective: how could two Lagrangians in possibly different Hamiltonian isotopy classes communicate with each other? This situation seems conceptually similar to the phenomenon of tunneling in quantum mechanics: there may be several minima of the classical potential function separated by potential wells, but there is a nontrivial quantum amplitude for the particles to move in between, thereby removing the ground state degeneracy. In the Thomas-Yau setup, we have two putative special Lagrangians, which are analogous to the energy minima points, and the Thomas-Yau uniqueness statement is similar to the removal of degeneracy, resulting in a unique ground state.

The physicists tell us that branes are dynamical objects and can fluctuate. If so, it might make sense to ask about the amplitude for a brane to start with a given configuration and end with another. Now if Lagrangian branes behave like classical particles moving on an infinite dimensional space such as Solomon’s space $\mathcal{O}$, one might expect Solomon’s formal picture to be relevant for describing this amplitude, and the incompleteness of $\mathcal{O}$ would suggest a nontrivial amplitude to tunnel outside $\mathcal{O}$ to another Hamiltonian isotopy class. As a conflicting viewpoint, the use of Floer theory in the Thomas-Yau argument suggests the Lagrangian branes communicate by the strings stretched between them, so one might expect the amplitudes to be computed in terms of worldsheet integrals. Lotay and Pacini’s geodesics fit this viewpoint better, and have the major advantage of making sense outside a given Hamiltonian isotopy class.

Tunneling effects made a famous appearance in Witten’s interpretation of Morse theory [80] in terms of supersymmetric quantum mechanics, with an eye towards applications in quantum field theory. In Witten’s context, the tunneling amplitude between two critical points of a Morse function is to leading order proportional to $e^{-S}$, where $S$ is proportional to the difference of the two critical values. Now Solomon’s functional has some similarity with a Morse function whose only critical points are minima. In sections 3.2, 3.7 we will unify the Solomon functional with the Lotay-Pacini geodesics, in the special case of exact Lagrangians. Could the Solomon functional have any physical interpretation in terms of the logarithm of the tunneling amplitudes? ⁴¹⁴¹ 41 As a grain of salt, Witten’s interpretation concerns supersymmetric QFT, while the Thomas-Yau picture concerns the dynamics of Lagrangian branes, which is a target space perspective on string theory, which lacks a fundamental path integral formulation. As such our suggested physcial interpretation is not logically rigorous, but only a guess.