1. The dynamics of space-time surfaces
This dynamics predicts two kinds of space-time regions (see this).
- The regions of first kind are locally minimal surfaces. These minimal surfaces are as 4-D analogs of geodesic lines analogs of asymptotic states of particle physics for which interactions are not on. They also satisfy non-linear geometrization of massless field equations so that both particle and wave aspects are present. What is especially important is that static minimal surfaces have vanishing mean curvature and look like saddles locally. They cannot be closed surface if stationary.
- Second type of regions are not minimal surfaces: there is a non-trivial coupling of the minimal surface term to 4-force density analogous to the divergence of Maxwellian energy momentum tensor. This is a generalization of the dynamics of a point-like charged particle in Maxwell field. These regions are identified as interaction regions: in particle physics these two regions correspond to external free particles and the interaction region. Magnetic flux tubes play fundamental role in TGD based quantum biology are deformations of string like objects, which represent simplest 4-D minimal surfaces.
Essential is the coupling between induced Kähler form (mathematically like Maxwell field) and the geometry of the surface: the divergence of energy momentum current assignable to the analog of cosmological term (4-volume) equals to the divergence of that assignable to Kähler action: this expresses local conservation of four-momentum. One could also speak about coupling between Kähler field and gravitational field: Penrose's intuition about the the role of gravitation in biology would be correct.
When the coupling is absent, minimal surface property implies the separate vanishing of both divergences and separate conservation of corresponding energy-momenta. All the known extremals of Kähler action are minimal surfaces: this is due to their very simple algebraic properties making easy to discover them. Physically this correspond to quantum criticality: dynamics is universal and does not depend on coupling parameters.
These observations lead to a rather general view about morphogenesis.
- The presence of the Kähler field (em field is sum of Kähler field and second term) makes possible flow equilibria such as cell membrane, which are not minimal surfaces. These surfaces can be closed and stationary making possible isolation from environment crucial for living organisms.
Spherical soap bubble is a good analogy: it is not minimal surface as the soap films spanned by frames are. They look locally like saddle surfaces with opposite external curvatures in two orthogonal directions, this implies that they cannot be closed surfaces. Bubble is not possible without a pressure difference Δ p between the interior and exterior of the bubble: the blowing of the soap bubble generates Δ p, and means external energy feed analogous to metabolic energy feed.
Δ p is analogous to a non-vanishing voltage V over cell membrane. The electric field of cell membrane and the energy feed providing the energy of electric field as metabolic energy are essential for the stability. More generally, V would generalize to non-vanishing of energy momentum tensor of Kähler field with non-vanishing divergence serving as a correlate for the energy transfer between Kähler and volume (gravitational) degrees of freedom.
This generalises to all morphologies, which correspond to closed surfaces. They necessarily involve both Kähler electric and magnetic fields coupling to the geometry to stabilize the morphology. This statement would give some content for the exaggerated claim that biology is nothing but electricity + Schrödinger equation that I heard during my first student year.
- For instance, the presence of Kähler electric field can correspond to electric fields of cell membrane or along a part of body. If it is too weak, things go wrong in development. As was found decades ago, consciousness is lost if the electric field between frontal lobes and hindbrain gets too weak or has wrong direction. Cell dies if the membrane potential becomes zero and EEG disappears in death. Also microtubules have electric field along their axes essential for their existence.
Michael Levin and his collaborators have discovered further fascinating connections between electric fields and morphogenesis. One of the discoveries is that the electric fields of the embryo are controlled by neurons of the still developing brain (see this). This conforms with the view that neurons and their MBs correspond to a higher level in the hierarchy than ordinary cells and there take care of control in longer scales. The MB of the developing brain would be the controller.
- A non-trivial coupling (four-momentum transfer) between the volume and Kähler degrees of freedom requires that the energy momentum currents have opposite and non-vanishing divergences. For the energy momentum tensor of ordinary Maxwell field the divergence is proportional to the contraction of Maxwell current and Maxwell field so that the current must be non-non-vanishing.
In TGD the energy momentum tensor is replaced with energy momentum current allowing to have well-defined notion of energy momentum and corresponding conservation laws. Now the divergence contains two terms. The first one is the contraction Tr(TKHk) of energy momentum tensor TK of Kähler action with the second fundamental form Hk: this term proportional to TK is new. Second term is proportional to the contraction jKJ∇ hk of the induced Kähler form J with Kähler current jK and gradients ∇ hk of imbedding space coordinates analogous the divergence of energy-momentum tensor jβFα~β in the case of ordinary Maxwell action. One expects both terms to be non-vanishing.
For the mere Kähler action, which I believed for decades to determine the preferred extremals, jK is either vanishing or light-like. In presence of coupling it can be both non-vanishing and time-like. The realization that cosmological term is present was forced by the twistor lift of TGD whose existence is possible only for H=M4× CP2.
- The predicted stabilizing Kähler (and em) currents would naturally correspond to the DC currents flowing along the body in various scales discovered already by Becker and found to be essential for the survival of the organism. In particular, Becker's DC currents are essential for the healing of wounds and in the regeneration of organs. In the first first aid stage of the healing DC currents are generated locally and after than central nervous system (CNS) takes care of the generation of the current (for TGD based discussion of Becker currents see this). Also this is easy to understand from the proposed stability criterion.
The emergence of life would require the coupling between Kähler and volume degrees of freedom. The following gives a quantitative discussion based on p-adic length scale hypothesis and twistor lift of TGD.
- The coefficient Λ/8π G == 1/L4 of the volume term in the action is analogous to cosmological constant in general relativity. The predicted wrong sign of Λ is the stumbling block of superstring theories. In TGD framework the sign is correct.
- p-Adic coupling constant evolution predicts that the cosmological constant depends on p-adic length scale L(k) characterizing the size scale of the Universe, most naturally as that of horizon size. In zero energy ontology (ZEO) L(k) is identifiable as the size scale of causal diamond (CD).
One important implication is a solution to the problem of cosmological constant. Although cosmological constant is huge at very early times (or more precisely, in very short p-adic length scales), it is small in the length scales of recent cosmology. The values of cosmological constant at smaller p-adic lengths scales are however visible also in the recent day physics in many-sheeted space-time and biology could make them visible as the following arguments show.
- There are two paired p-adic length scales: short p-adic length scale L(k1) and long p-adic length scale L(k). The vacuum energy density ρvac=Λ/8π G is naturally proportional to 1/L4(k1). One has energy E=1/L(k1) per 3-volume L(k1)3.
ρvac=Λ/8π G is also naturally proportional to 1/GL2(k) since Λ =x/L(k)2 is natural by dimensional considerations. If L(k) corresponds to the size scale of the horizon, Λ degreases during cosmic evolution and the problem of cosmological constant disappears. One has
1/L4(k1) = Λ/8π G ,
Λ/8π = x2/L2(k) .
Here the p-adic length scale L(k) could characterize the p-adic size scale of CD. G=lPl2 is gravitational constant, lPl Planck length scale, and L=L(k1) is a smaller length scale. L(k1) expressible using the geometric mean
L(k1)=(8π GΛ)1/4 = x-1/2(L(k)lPl)1/2 .
of L(k) and Planck length lPl and allows an identification as a p-adic length scale for a suitable choices of the parameter x. One has (8π)1/4≈ 2.4.
What could this pairing of short and long p-adic length scales mean? The notion of magnetic body (MB) could provide and explanation. MB has onion-like layered structure with layers labelled by p-adic length scales up to some maximum size scale. This suggests that a biological structure with size scale L(k1) has MB for which the largest layer has the size scale L(k). L(k1) would correspond to smallest length scale in the hierarchy. Both scales could correspond to size scales of CDs.
Remark: When L(k1) is scaled by 2r (k1→ k1+r), L(k) is scaled by 22r (k→ k+2r).
- From the parameterization
ρvac= yH2/8π G
of the dark energy density in terms of Hubble constant at given space-time sheets one obtains an estimate for the inverse of the Hubble constant H, which depends on space-time sheet in terms of L(k), as
1/H(k) = (y/8π x)1/2 L(k) .
H(k) refers now to Hubble constant in given p-adic length scale characterizing a level in the hierarchy of space-time sheets and is not the ordinary Hubble constant defined in very long scales at GRT limit of TGD. Naturality suggests the condition y/8π x=1.
- To achieve this, Kähler action ∝ E2-B2 must be of the same order of magnitude as (Λ/8π G)== x/GL2(k) giving in the case of cell membrane for the Kähler electric field strength the rough estimate
E ∼ (x/lPlL(k))1/2 .
Remark: The electric field of the cell membrane corresponds to E ∼ 5× 10-4 eV2 in the units of particle physicist (hbar=1 and c=1) in which unit of distance is 1/eV and one has 1 m ↔ 1.24× 106 eV-1.
- If an estimate for the typical strength E of bio-electric field is given, one can get some idea about the length scale L(k) as
L(k)= x/lPlE .
By feeding in Planck length lPl∼ 1.6× 10-35 m and the electric field E∼ 5× 106 V/m of the cell membrane, one obtains for the cell membrane the estimate
L(k)∼ x1/2 L0 ,
L0= 1.1× 106 ly .
L(k1)= x-1/4 L1 ,
L1=(lPlL0)1/2=4.2× 10-7 m .
Note that L(k) scales as x1/2 and L(k1) as x-1/4.
- The value of electric field for cell membrane is essential for the argument. If one wants to generalize the argument from cell membrane to other systems, one must have an idea about how it scales. Membrane potential is near the value for which the potential energy ZeV0 for a Cooper pair is slightly above the thermal energy at physiological temperature. Hence the possible magnetic flux tube assignable to membrane proteins acting as Josephson junctions through cell membrane carry weakest possible electric field: this conforms with metabolic economy. A natural generalization would be that for a flux tube of length L one has E= V0/L. This gives the scalings
L(k)∝ L/Lc ,
L(k1)∝ (L/Lc)1/2 .
Option I: The size of the Milky Way is estimated to be about LMW=105 ly. L(k)=LMW would be obtained for x=.01. One should be however cautious with this estimate: also x∼ 1 might be acceptable.
- For L(k1) the formula L(k1)= x-1/2(L(k)lPl)1/2 gives for x=.01
L(k1)= 4 nm .
This is near the p-adic length scale L(149)=5 nm assignable to the ordinary cell membrane. There are indeed indications that galactic year defines a biorhythm. For x=1 giving L(k)= 106 ly one would have
L(k1)= 1.26 nm, which does not correspond to cell membrane length scale.
- For the inverse of the Hubble constant H(149) one obtains for x=.01 the estimate
1/H(k) ≈ 2(y/8π x)1/2 L(k) .
H(149) does not correspond to standard cosmological constant. One has H(149)=L(k) for y=2π x=.0628.
- The scaling L(k) → 105L(k) the size scale of the observed Universe about 15 Gly scales L(k1=149) to L(k1)= 1.3 μm, which corresponds to L(165)=1.25 μm in a reasonable approximation (L(167)=2.5 μm is the p-adic length scale of nuclear membrane). This scale would correspond to a distance through which one has membrane potential V0. Could the size scales of galaxy and observed Universe indeed correspond to those of lipid layer of cell membrane and cell membrane?
- The condition that L(k)=RE=6.3× 106 m gives x=6.4× 10-16 and L(k1)= 6.7 mm. L(k1) could characterize a brain structure involved in the generation of EEG. Note that the estimate assumes the electric field of cell membrane. One can argue that the value of x=6.4× 10-16 is highly un-natural.
- There are indications for the existence of life in Mars, whose radius is 1/2 of that for Earth. L(k) would scale down by 1/2 as also the cell membrane thickness. Could this be assumed also for the Option I? By the proposed criterion the strength of electric field E for cell membrane should be 2 times stronger than for Earthly cell (for same physiological temperature). For instance, membrane potential could be same but membrane thickness could be 1/2 of that for Earthly membrane.
Interestingly, the TGD based version of Expanding Earth model predicts that Earth experienced a rapid expansion doubling its radius. Even more, neuronal cell membranes are 2 times thicker than ordinary cell membranes. Animals utilizing aerobic respiration emerged in Cambrian explosion and eventually also neurons and TGD suggests an explanation in terms of oxygenation as the life in underground oceans entered to the surface through the cracks generated by the expansion.
See also the forthcoming article Getting philosophical: some comments about the problems of physics, neuroscience, and biology.
For a summary of earlier postings see Latest progress in TGD.