Quantum tunneling means that quantum particle is able to move through a classically forbidden region, where its momentum would be imaginary. The tunnelling probability can be estimated by solving the Schrödinger equation assuming that a free particle described as a wave arrives from the other side of the barrier and is partially reflected and partially transmitted. Tunneling probability is proportional to exp(-2∫ Kdx), k=iK is the imaginary wave vector in forbidden region - imaginary because the kinetic energy T=p2/2m of particle equals to T= E-V and is negative. In forbidden region momentum p is imaginary as also the wave vector k=iK = p/hbar. The trasmission-/tunnelling probability decreases exponentially with the height and width of the barrier. Hence the tunnelling should be extremely improbable in macroscopic and even nano-scales. The belief has been that this is true also in chemistry. Especially so at high temperatures, where quantum coherence lengths are expected to be short. Experiments have forced to challenge this belief.
In TGD framework the hierarchy of phases of ordinary matter with Planck constant given by heff=n× h. The exponent in the tunneling probablity is proportional to 1/hbar. If hbar is large, the tunnelling probability increases since the damping exponential is near to unity. Tunneling becomes possible in scales, which are by a factor heff/h=n longer than usually. At microscopic level - in the sense of TGD space-time - the tunnelling would occur along magnetic flux tubes. This could explain the claimed tunneling effects in chemistry. In biochemistry these effects would of special importance.
In TGD framework non-standard values of Planck constant are associated with quantum criticality and there is experimental evidence for quantum criticality in the bio-chemistry of proteins (see also this. In TGD framework quantum criticality is the basic postulate about quantum dynamics in all length scales and makes TGD unique since the fundamental coupling strength is analogous to critical temperature and therefore has a discrete spectrum.
Physics student reading this has probably already noticed that diffraction is another fundamental quantum effect. By naive dimensional estimate, the sizes of diffraction spots should scale up by heff. This might provide a second manner to detect the presence of large heff photons and also other particles such as electrons. Dark variants of particles wold not be directly observable but might induce effects in ordinary matter making the scaled up diffraction spots visible. For instance, could our visual experience provide some support for large heff diffraction? The transformation of dark photons to biophotons might make this possible.
P. S. Large heff quantum tunnelling could provide one further mechanism for cold fusion. The tunnelling probabily for overcoming Coulomb wall separating incoming charged nucleus from target nucleus is extremely small. If the value of Planck constant is scaled up, the probability increases by the above mechanism. Therefore TGD allows to consider at least 3 different mechanisms for cold fusion: all of them would rely on hierarchy of Planck constants.
For a summary of earlier postings see Latest progress in TGD.