Wednesday, February 10, 2016

Cosmic evolution of the radius of the fiber of the twistor space of space-time surface

I have continued the little calculations inspired by the surprising finding that twistorial lift of Kähler action based dynamics immediately leads to the identification of cosmological length scales as fundamental classical length scales appearing in 6-D Kähler action, whose dimensional reduction gives Kähler action plus small cosmological term with correct sign to explain together with magnetic flux tube tension accelerating cosmic expansion. Whether Planck length emerges classically from from quantum theory remains still an open question.

For a fleeting moment I thought that for the twistor space of Minkowski space the 2-D fiber could be hyperbolic sphere H2 (t2-x2-y2 =-RH2) rather than sphere S2 as it is for CP2 with Euclidian signature of metric. I however soon realized that the infinite area of H2 implies that 6-D Kähler action is infinite and that there are many other difficulties.

The correct manner to define Minkowskian variant of twistor space is by starting from the generalization of complex and Kähler structures for M4= M2+ E2 of local tangent space to longitudinal (defined by light-like vector) and to transversal directions (polarizations orthogonal to the light-like vector. The decomposition can depend on point but the distributions of two planes must integrated to 2-D surfaces. In E2 one has complex structure and in M2 its hyper-complex variant. In M2 has decomposition of replacing complex numbers by hyper-complex numbers so that complex coordinate x+iy is replaced with w=t+ie, i2=-1 and e2=-1.

It took time to realize I have actually carried out this generalization years ago with quite different motivations and called the resulting structure Hamilton-Jacobi structure! The twistor fiber is defined by projections of 4-D antisymmetric tensors (in particular induced Kähler form) to the orthogonal complement of unique time direction determed by the sum of light-like vector and its dual in M2. This part of tensor could be called magnetic. Th magnetic part of the tensor defines a direction and one has natural metric making the space of directions sphere S2 with metric having signature (-1,-1). This requires that twistor space has metric signature (-1,-1,1,-1,-1,-1) (I also considered seriously the signature (1,1,1,-1,-1,-1) so that there are three time-like coordinates) .

The radii of the spheres associated with M4 and CP2 define two fundamental scales and the scaling of 6-D Käler action brings in third fundamental length scale. On possibility is that the radii of the two spheres are actually identical and essentially equal to CP2 radius. Second option is that the radius of S2(M4) equals to Planck length, which would be therefore a fundamental length scale.

The radius RD of the 2-D fiber of twistor space assignable to space-time surfaces is dynamical. In Euclidian space-time regions the fiber is sphere: a good guess is that its order of magnitude is determined by the winding numbers of the maps from S2(X4)→ S2(M4) and S2(X4) → S2(CP2). The winding numbers (1,0) and (0,1) represent the simplest options. The question is whether one could say something non-trivial about cosmic evolution of RD as function of cosmic time. This seems to be the case.

One can actually get estimate for the evolution of RD as function of cosmic time if one accepts Friedman cosmology as an approximation of TGD cosmology.

  1. Assume critical mass density so that one has

    ρcr= 3H2/8π G .


  2. Assume that the contribution of cosmological constant term to the mass mass density dominates. This gives ρ≈ ρvac=Λ/8π G. From ρcrvac one obtains

    Λ= 3H2 .

  3. From Friedman equations one has H2= ((da/dt)/a)2, where a corresponds to light-cone proper time and t to cosmic time defined as proper time along geodesic lines of space-time surface approximated as Friedmanncosmology. One has

    Λ= 3/gaaa2

    in Robertson-Walker cosmology with ds2= gaada2-a232.

  4. Combining this equations with the TGD based equation

    Λ= 8π2G/L2RD2

    one obtains

    2G/L2RD2= 3/gaaa2.

  5. Assume that quantum criticality applies so that L has spectrum given by p-adic length scale hypothesis so that one discrete p-adic length scale evolution for the values of L. There are two options to consider depending on whether p-adic length scales are assigned with light-cone proper time a or with cosmic time t

    T= a (Option I) , T=t (Option II).

    Both options give the same general formula for the p-adic evolution of L(k) but with different interpretation of T(k).

    L(k)/Lnow= T(k)/Tnow , T(k)= L(k) = 2(k-151)/2× L(151) , L(151)≈ 10 nm .

    Here T(k) is assumed to correspond to primary p-adic length scale. An alternative - less plausible - option is that T(k) corresponds to secondary p-adic length scale L2(k)=2k/2L(k) so that T(k) would correspond to the size scale of causal diamond. In any case one has L ∝ L(k). One has a discretized version of smooth evolution

    L(a) = Lnow × (T/Tnow) .

Consider now the predictions.
  1. Feeding into the formula following from two expressions for Λ one obtains an expression for RD(a)

    RD/lP= (8/3)1/2π× (a/L(a)× gaa1/2

    This equation tells that RD is indeed dynamical, and becomes very small at very early times since gaa becomes very small. As a matter of fact, in very early cosmic string dominated cosmology gaa would be extremely small constant (see this). In late cosmology gaa→ 1 holds true and one obtains at this limit

    RD(now)= (8/3)1/2π× (anow/Lnow) × lP ≈ 4.4 ×(anow/Lnow) × lP .

  2. For T= t option RD/lP remains constant during both matter dominated cosmology, radiation dominated cosmology, and string dominated cosmology since one has a∝ tn with n= 1/2 during radiation dominated era, n= 2/3 during matter dominated era, and n=1 during string dominated era (see this). This gives

    RD/lP=(8/3)1/2π× at (gaa1/2(t(end)/L(end)) = (8/3)1/2π×(1/n)(t(end)/L(end)) .

    Here "end"> refers the end of the string or radiation dominated period or to the recent time in the case of matter dominated era. The value of n would have evolved as RD/lP∝ (1/n (tend/Lend), n∈ [1,3/2,2}. During radiation dominated cosmology RD ∝ a1/2 holds true. The value of RD would be very nearly equal to R(M4) and R(M4) would be of the same order of magnitude as Planck length. In matter dominated cosmology would would have RD ≈ 2.2 (t(now)/L(now)) × lP .

  3. For RD(now)=lP one would have

    Lnow/anow =(8/3)1/2π≈ 4.4 .

    In matter dominated cosmology gaa=1 gives tnow=(2/3)× anow so that predictions differ only by this factor for options I and II. The winding number for the map S2(X4)→ S2(CP2) must clearly vanish since otherwise the radius would be of order R.

  4. For RD(now)= R one would obtain

    anow/Lnow =(8/3)1/2π× (R/lP)≈ 2.1× 104 .

    One has Lnow=106 ly: this is roughly the average distance scale between galaxies. The size of Milky Way is in the range 1-1.8 × 105 ly and of an order of magnitude smaller.

  5. An interesting possibility is that RD(a) evolves from RD ∼ R(M4) ∼ lP to RD ∼ R. This could happen if the winding number pair (w1,w2)=(1,0) transforms to (w1,w2)=(0,1) during transition to from radiation (string) dominance to matter (radiation) dominance. RD/lP radiation dominated cosmology would be related by a factor

    RD(rad)/RD(mat)= (3/4)(t(rad,end)/L(rad,end))×(L(now)/t(now))

    to that in matter dominated cosmology. Similar factor would relate the values of RD/lP in string dominated and radiation dominated cosmologies. The condition RD(rad)/RD(mat) =lP/R expressing the transformation of winding numbers would give

    L(now)/L(rad,end) =(4/3) (lP/R) (t(now)/t(rad,end)) .

    One has t(now)/t(rad,end)≈ .5× 106 and lP/R =2.5× 10-4 giving L(now)/L(rad,end)≈ 125, which happens to be near fine structure constant.

R(M4)∼ lP seems rather plausible option so that Planck length would be fundamental classical length scale emerging naturally in twistor approach. Cosmological constant would be coupling constant like parameter with a spectrum of critical values given by p-adic length scales.

See the the article From Principles to Diagrams or the chapter TGD Variant of Twistor Story of "Towards M-matrix".

For a summary of earlier postings see Links to the latest progress in TGD.

Monday, February 08, 2016

Twistors and the relationship of TGD to GRT

For year or two ago I ended up with a vision about how twistor approach could generalize to TGD framework. A more explicit realization of twistorialization as lifting of the preferred extremal X4 of Kähler action to corresponding 6-D twistor space X6 identified as surface in the 12-D product of twistor spaces of M4 and CP2 allowing Kähler structure suggests itself: this makes these spaces completely unique twistorially and seems more or less obvious that the Kähler structure must have profound physical meaning. It turned out that it has: the projection of Kähler form defines the representation of preferred quaternionic imaginary unit needed to assign twistor structure to space-time surface. Almost equally obvious idea is that the lifting of the dynamics for space-time surface to that for its twistor space in the product of twistor spaces of M4 and CP2 must be based on 6-D Kähler action.

Contrary to the original expectations, the twistorial approach is not mere reformulation but leads to a first principle identification of cosmological constant and perhaps also of gravitational constant and to a modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure.

There are some new results forcing a profound modification of the recent view about TGD but consistent with the general picture. A more explicit realization of twistorialization as lifting of the preferred extremal X4 of Kähler action to corresponding 6-D twistor space X6 identified as surface in the 12-D product of twistor spaces of M4 and CP2 allowing Kähler structure suggests itself.

The action principle in 6-D context is also Kähler action, which dimensionally reduces to Kähler action plus cosmological term. This brings in the radii of spheres S2 associated with the twistor space of CP2 presumably determined by CP2 radius and radius of S2 associated with M4 twistor space for which an attractive identification is as Planck length, which would be now purely classical parameter. The radius of S2 associated with space-time surface is determined by induced metric and is emergent length scale. The normalization of 6-D Kähler action by a scale factor with dimension which is inverse length squared brings in a further length scale closely related to cosmological constant which is also dynamical and has correct sign to explain accelerated expansion of the Universe.

The dimensionally reduced dynamics is a highly non-trivial modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure.

In the sequel I will discuss the recent understanding of twistorizalization, which is considerably improved from that in the earlier formulation. I formulate the dimensional reduction of 6-D Kähler action and consider the physical interpretation. After that I proceed to discuss the basic principles behind the recent view about scattering amplitudes as generalized Feynman diagrams.

1. Some mathematical background

First I will try to clarify the mathematical details related to the twistor spaces and how they emerge in the recent context. I do not regard myself as a mathematician in technical sense and I can only hope that the representation based on physical intuition does not contain serious mistakes.

1.1. Imbedding space is twistorially unique

It took roughly 36 years to learn that M4 and CP2 are twistorially unique. Space-times are surfaces in H=M4× CP2. M4 and CP2 are unique 4-manifolds in the sense that both allow twistor space with Kähler structure: Kähler structure is the crucial concept. Strictly speaking, M4 and its Euclidian variant E4 allow both twistor space and the twistor space of M4 is Minkowskian variant T(M4)= SU(2,2)/SU(2,1)× U(1) of 6-D twistor space CP3= SU(4)/SU(3)× U(1) of E4. The twistor space of CP2 is 6-D T(CP2)= SU(3)/U(1)× U(1), the space for the choices of quantization axes of color hypercharge and isospin.

This leads to a proposal - just a proposal - for the formulation of TGD in which space-time surfaces X4 in H are lifted to twistor spaces X6, which are sphere bundles over X4 and such that they are surfaces in 12-D product space T(M4)× T(CP2) such the twistor structure of X4 are in some sense induced from that of T(M4)× T(CP2). What is nice in this formulation is that one can use all the machinery of algebraic geometry so powerful in superstring theory (Calabi-Yau manifolds).

1.2 What does twistor structure in Minkowskian signature really mean?

What twistor structure in Minkowskian signature really means geometrically has remained a confusing question for me. The problems associated with the Minkowskian signature of the metric are encountered also in twistor Grassmann approach to scattering amplitudes but are circumvented by performing Wick rotation that is using E4 or S4 instead of M4 and applying algebraic continuation. Also complexification of Minkowksi space for momenta is used. These tricks do not apply now.

Let us try to collect thoughts about what is involved.

  1. Instead of M4 one considers the conformal compactification M4c of M4 identifiable as the boundary of light-cone boundary of 6-D Minkowski space with signature (1,1,-1,-1,-1), whose points differing by scaling are identified. One has a slicing by spheres of signature (-1,-1,-1) and varying radius ρ and these spheres are projectively identified so that one can "fix the gauge" by choosing ρ=ρ0. Since one has light-cone, the contribution dρ2 to the line element vanishes and one obtains ds2= ρ022- ρ20 ds2(S3). Conformal compactification means that the scale ρ0 of the metric is not unique. The scaling of the metric of the twistor space ρ02. Conformal invariance of the theory saves from problems.

  2. The Euclidian version of the twistor space of M4 corresponds to the twistor space of S4 identifiable as CP3= SU(4)/SU(3)× U(1) identifiable in terms of complex 2+2-spinors. The twistor space of M4c is SU(2,2)/SU(2,1)× U(1) (see this) and can be seen as a kind of algebraic continuation of CP3=SU(4)/SU(3)× U(1). This space is complex manifold but it is not completely clear to me whether this really guarantees the existence of Kähler structure consistent with the complex structure.

  3. If the Minkowskian variant of the twistor space (rather than only that associated with S4) is to have complex structure in the ordinary sense of the word, its metric must have even signature. M4c has signature (1,-1-1,-1) so that the signature of the analog of S2 fiber should have signature which is (1,-1) to give even signature (1,-1,1-1,-1,-1) for the twistor space. The sphere S2 would be replaced with its non-compact hyperbolic counterpart SO(2,1)/SO(1,1) and has metric signature (1,-1). One cannot assign to it finite size in the usual sense. However, since this space corresponds to hyperboloid t2-x2-y2=-R2 (3-D mass shell), one can assign to it finite hyperbolic radius RH. There are however problems.

    1. One cannot assign to H2 finite size in the usual sense. However, since this space corresponds to hyperboloid t2-x2-y2=-R2 (3-D mass shell), one could assign to it finite hyperbolic radius RH. In dimensional reduction of 6-D Kähler action however the integral over H2 gives its area if the restriction of J to H2 has square equal to metric as is extremely natural to assume. The area is RH2 times an infinite number and 4-D dimensionally reduced action would have infinite value. At the limit RH=0 (2-D light-cone boundary) the area vanishes as also the dimensionally reduced action.

    2. For even signature of twistor space the determinant of the induced 6-metric would be real in both Euclidian and Minkowskian space-time regions. Both Euclidian Minkowskian regions contribute to Kähler function (as was the original wrong assumption using |det(g)1/2| in volume element). The exponent of Kähler action in Minkowskian regions would not define phase as QFT picture demands.

    3. This picture is in conflict with the vision about the fiber as space S2 of directions defined by antisymmetric forms. The hidden assumption is that one has field of preferred time-like directions n and one considers induced Kähler form at space-like 3-surface with metric signature (-1,-1,-1) with n as time-like normal field.

      Could one imagine fixing of space-like direction field defining normals for a slicing by 3-surfaces with metric signature (1,-1,-1)? If so, one would end up with SO(2,1)/SO(1,1) as the fiber characterizing directions of projections of J to this subspace. The slicing by 3-surfaces parallel to the light-like 3-surfaces at the boundaries of Minkowskian and Euclidian space-time regions could indeed do the job. The light-likeness of these 3-surfaces also fits nicely with conformal invariance. The above problems are however enough to guarantee that the lifetime of H2 option was rather precisely 24 hours.

  4. The only alternative, which comes in mind is a hyperbolic generalization of the Kähler structure. This requires that the metric of the Minkowskian twistor space has signature (1,1,1,-1,-1,-1). This would give 3 time-like directions and each hypercomplex coordinate would correspond to a sub-spaces with signature (1,-1). Hypercomplex coordinates can be represented as h=t+iez, i2=-1,e2=-1. Kähler form representing imaginary unit would be replaced with eJ. One would consider sub-spaces of complexified quaternions spanned by real unit and units eIk, k=1,2,3 as representation of the tangent space of space-time surfaces in Minkowskian regions. This is familiar already from M8 duality (see this).

    One could regard Minkowskian twistor space as a kind of Wick rotation of the Euclidian twistor space. Hyper-complex numbers do not define number field since for light-like hypercomplex numbers t+iez, t=+/- z do not have finite inverse. Hypercomplex numbers allow a generalization of analytic functions used routinely in physics. Fiber would be sphere S2 with metric signature (1,1). Cosmological term would be finite and the sign of the cosmological term in the dimensionally reduced action would be positive as required. Also metric determinant would be imaginary as required. At this moment I cannot invent any killer objection against this option.

1.3 What the induction of twistor structure could mean?

To proceed one must make explicit the definition of twistor space. The 2-D fiber S2 consists of antisymmetric tensors of X4 which can be taken to be self-dual or anti-self-dual by taking any antisymmetric form and by adding to its plus/minus its dual. Each tensor of this kind defines a direction - point of S2. These points can be also regarded as quaternionic imaginary units. One has a natural metric in S2 defined by the X4 inner product for antisymmetric tensors: this inner product depends on space-time metric. Kähler action density is example of a norm defined by this inner product in the special case that the antisymmetric tensor is induced Kähler form. Induced Kähler form defines a preferred imaginary unit and is needed to define the imaginary part ω(X,Y)= ig(X,-JY) of hermitian form h= h+iω.

Consider now what the induction of twistor structure could mean.

  1. The induction procedure for Kähler structure of 12-D twistor space T requires that the induced metric and Kähler form of the base space X4 of X6 obtained from T is the same as that obtained by inducing from H=M4× CP2. Since the Kähler structure and metric of T is lift from H this seems obvious. Projection would compensate the lift.

  2. This is not yet enough. The Kähler structure and metric of F projected from T must be same as those lifted from X4. The connection between metric and ω implies that this condition for Kähler form is enough. The antisymmetric Kähler forms in fiber obtained in these two manners co-incide. Since Kähler form has only one component in 2-D case, one obtains single constraint condition giving a commutative diagram stating that the direct projection to F equals with the projection to the base followed by a lift to fiber. The resulting induced Kähler form is not covariantly constant but in fiber F one has J2=-g.

    As a matter of fact, this condition might be trivially satisfied as a consequence of the bundle structure of twistor space. The Kähler form from S2× S2 can be projected to S2 associated with X4 and by bundle projection to a two-form in X4. The intuitive guess - which might be of course wrong - is that this 2-form must be same as that obtained by projecting the Kähler form of CP2 to X4. If so then the bundle structure would be essential but what does it really mean?

  3. Intuitively it seems clear that X6 must decompose locally to a product X4× S2 in some sense. This is true if the metric and Kähler form reduce to direct sums of contributions from the tangent spaces of X4 and S2. This guarantees that 6-D Kähler action decomposes to a sum of 4-D Kähler action and Kähler action for S2.

    This could be however too strong a condition. Dimensional reduction occurs in Kaluza-Klein theories and in this case the metric can have also components between tangent spaces of the fiber and base being interpreted as gauge potentials. This suggests that one should formulate the condition in terms of the matrix T↔ gαμgβν-gανgβμ defining the norm of the induced Kähler form giving rise to Kähler action. T maps Kähler form J↔ Jαβ to a contravariant tensor Jc↔ Jαβ and should have the property that Jc(X4) (Jc( S2)) does not depend on J( S2) (J(X4)).

    One should take into account also the self-duality of the form defining the imaginary unit. In X4 the form S=J+/- *J is self-dual/anti-self dual and would define twistorial imaginary unit since its square equals to -g representing the negative of the real unit. This would suggest that 4-D Kähler action is effectively replaced with (J+/- *J)∧(J+/- *J)/2 =J*∧J +/- J∧J, where *J is the Hodge dual defined in terms of 4-D permutation tensor ε. The second term is topological term (Abelian instanton term) and does not contribute to field equations. This in turn would mean that it is the tensor T+/- ε for which one can demand that Sc(X4) (Sc(S2)) does not depend on S(S2) (S(X4)).

2. Surprise: twistorial dynamics does not reduce to a trivial reformulation of the dynamics of Kähler action

I have thought that twistorialization classically means only an alternative formulation of TGD. This is definitely not the case as the explicit study demonstrated. Twistor formulation of TGD is in terms of of 6-D twistor spaces T(X4) of space-time surfaces X4⊂ M4× CP2 in 12-dimensional product T=T(M4)× T(CP2) of 6-D twistor spaces of T(M4) of M4 and T(CP2) of CP2. The induced Kähler form in X4 defines the quaternionic imaginary unit defining twistor structure: how stupid that I realized it only now! I experienced during single night many other "How stupid I have been" experiences.

Classical dynamics is determined by 6-D variant of Kähler action with coefficient 1/L2 having dimensions of inverse length squared. Since twistor space is bundle, a dimensional reduction of 6-D Kähler action to 4-D Kähler action plus a term analogous to cosmological term - space-time volume - takes place so that dynamics reduces to 4-D dynamics also now. Here one must be careful: this happens provided the radius of F associated with X4 does not depend on point of X4. The emergence of cosmological term was however completely unexpected: again "How stupid I have been" experience. The scales of the spheres and the condition that the 6-D action is dimensionless bring in 3 fundamental length scales!

2.1 New scales emerge

The twistorial dynamics gives to several new scales with rather obvious interpretation. The new fundamental constants that emerge are the radius of hyperbolic sphere associated with T(M4) and of sphere associated with T(CP2). The radius of the fiber associated with X4 is not a fundamental constant but determined by the induced metric. By above argument the fiber is sphere for Euclidian signature and hyperbolic sphere for Minkowskian signature.

  1. For CP2 twistor space the radius of S2 must be apart from numerical constant equal to CP2 radius R. For M4 the simplest assumption is that also now the radius for S2(M4 equals to R(M4=R so that Planck length would not emerge from fundamental theory classically. Second option is that it does and one has R(M4=lP.

  2. For space-time regions with Minkowskian/Euclidian signature S2(X4) has signature (1,1)/(-1,-1). In Minkowskian regions/Euclidian regions the contribution from S2(M4)/S2(CP2) to the induced metric at fiber dominates. The radius RD of S2(X4) is dynamically determined.


Consider now the possible interpretation of these length scales and secondary scales derived from them. First some general comments are in order.
  1. One can deduce an expression for cosmological constant Λ and show that it is positive. 6-D Kähler action has dimensions of length squared and one must scale it by a dimensional constant: call it 1/L2. L is a fundamental scale and in dimensional reduction it gives rise to cosmological constant. Cosmological constant Λ is defined in terms of vacuum energy density as Λ =8π Gρvac can have two interpretations. Λ can correspond to a modification of Einstein-Hilbert action or - as now - to an additional term in the action for matter. In the latter case positive Λ means negative pressure explaining the observed accelerating expansion. It is actually easy to deduce the sign of Λ.

    1/L2 multiplies both Kähler action - FijFij (∝ E2-B2 in Minkowskian signature). The energy density is positive. For Kähler action the sign of the multiplier must be positive so that 1/L2 is positive. The volume term is fiber space part of action having same form as Kähler action. It gives a positive contribution to the energy density and negative contribution to the pressure.

    In Λ= 8π Gρvac one would have ρvac=π/L2RD2 as integral of the -FijFij over S2 given the π/RD2 (no guarantee about correctness of numerical constants). This gives Λ= 8π2G/L2RD2. Λ is positive and the sign is same as as required by accelerated cosmic expansion. Note that super string models predict wrong sign for Λ. Λ is also dynamical since it depends on RD, which is dynamical. One has 1/L2 =kΛ, k=8π2G/RD2 apart from numerical factors.

    The value of L of deduced from Euclidian and Minkowskian regions in this formal manner need not be same. Since the GRT limit of TGD describes space-time sheets with Minkowskian signature, the formula seems to be applicable only in Minkowskian regions. Again one can argue that one cannot exclude Euclidian space-time sheets of even macroscopic size and blackholes and even ordinary concept matter would represent this kind of structures.

  2. L is not size scale of any fundamental geometric object. This suggests that L is analogous to αK and has value spectrum dictated by p-adic length scale hypothesis. In fact, one can introduce the ratio of ε=R2/L2 as a dimensionless parameter analogous to coupling strength what it indeed is in field equations. If so, L could have different values in Minkowskian and Euclidian regions.

  3. I have earlier proposed that RU==(1/Λ)1/2 is essentially the p-adic length scale Lp ∝ p1/2= 2k/2, p≈ 2k, k prime, characterizing the cosmology at given time and satisfies RU∝ a meaning that vacuum energy density is piecewise constant but on the average decreases as 1/a2, a cosmic time defined by light-cone proper time. A more natural hypothesis is that L satisfies this condition and in turn implies similar behavior or RU. p-Adic length scales would be the critical values of L so that also p-adic length scale hypothesis would emerge from quantum critical dynamics! This conforms with the hypothesis about the value spectrum of αK labelled in the same manner (see this).

  4. At GRT limit the magnetic energy of the flux tubes gives rise to an average contribution to energy momentum tensor, which effectively corresponds to negative pressure for which the expansion of the Universe accelerates. It would seem that both contributions could explain accelerating expansion. If the dynamics for Kähler action and volume term are coupled, one would expect same orders of magnitude for negative pressure and energy density - kind of equipartition of energy.

Consider first the scales emerging from GRT picture. RU ∼ (1/Λ1/2∼ 1026 m = 10 Gly is not far from the recent size of the Universe defined as c× t ∼ 13.8 Gly. The derived size scale L1==(RU× lP)1/2 is of the order of L1=.5× 10-4 meters, the size of neuron. Perhaps this is not an accident. To make life of the reader easier I have collected the basic numbers to the following table.

m(CP2)≈ 5.7× 1014 GeV mP=2.435 × 1018 GeV R(CP2)/lP≈ 4.1× 103
RU= 10 Gy t= 13.8 Gy L1= (lPRU)1/2=.5 × 10-4

Let us consider now some quantitative estimates. One can consider several limiting cases.

  1. M4 contribution to the metric of S2(X4) dominates and one has R(X4)=R(M4). The naive first guess is R(M4)=lP so that Planck scale would be fundamental length. R(M4)=R is second good motivated by the expectation that Planck length involves Planck constant and is quantum scale. Planck length scale could emerge in more complex manner as has indeed been the belief hitherto. For Euclidian regions the radius of sphere would satisfy RD≤ R (CP2 size).

  2. For R(M4)=R one can imagine that RD is reduced to lP dynamically as the contributions from the fibers almost cancel each other.
Consider next additional scales emerging from TGD picture.
  1. One has L = ( 23/2π lP/RD)× RU. In Minkowskian regions with RH=lP this would give L = 8.9× RU: there is no obvious interpretation for this number. If one takes the formula seriously in Euclidian regions one obtains the estimate L=29 Mly. The size scale of large voids varies from about 36 Mly to 450 Mly (see this).

  2. Consider next the derived size scale L2=(L× lP)1/2 = [L/RU]1/2 × L1 = [23/2π lP/RD]1/2× L1. For RD=lP one has L2 ≈ 3L1. For RD=R making sense in Euclidian regions, this is of the order of size of neutrino Compton length: 3 μm, the size of cellular nucleus and rather near to the p-adic length scale L(167)= 2.6 m, corresponds to the largest miracle Gaussian Mersennes associated with k=151,157,163,167 defining length scales in the range between cell membrane thickness and the size of cellular nucleus. Perhaps these are co-incidences are not accidental. Biology is something so fundamental that fundamental length scale of biology should appear in the fundamental physics.
The formulas and predictions for different options are summarized by the following table.

Option L=[23/2π lP/RD]× RU L2=(LlP)1/2 = [23/2π lP/RD]1/2× L1
RD= R 29 Mly ≈ 3 μ m
RD=lP 8.9RU ≈ 3L1=1.5× 10-4 m

In the case of M4 the radius of S2 cannot be fixed it remains unclear whether Planck length scale is fundamental constant or whether it emerges.

2.2 What about extremals of the dimensionally reduced 6-D Kähler action?

It seems that the basic wisdom about extremals of Kähler action remains unaffected and the motivations for WCW are not lost. What is new is that the removal of vacuum degeneracy is forced by twistorial action.

  1. All extremals, which are either vacuum extremals or minimal surfaces remain extremals. In fact, all extremals that I know. For minimal surfaces the dynamics of the volume term and 4-D Kähler action separate and field equations for them are separately satisfied. The vacuum degeneracy motivating the introduction of WCW is preserved. The induced Kähler form vanishes for vacuum extremals and the imaginary unit of twistor space is ill-defined. Hence vacuum extremals cannot belong to WCW. This correspond to the vanishing of WCW metric for vacuum extremals.

  2. For non-minimal surfaces Kähler coupling strength does not disappear from the field equations and appears as a genuine coupling very much like in classical field theories. Minimal surface equations are a generalization of wave equation and Kähler action would define analogs of source terms. Field equations would state that the total isometry currents are conserved. It is not clear whether other than minimal surfaces are possible, I have even conjectured that all preferred extremals are always minimal surfaces having the property that being holomorphic they are almost universal extremals for general coordinate invariant actions.

  3. Thermodynamical analogy might help in the attempts to interpret. Quantum TGD in zero energy ontology (ZEO) corresponds formally to a complex square root of thermodynamics. Kähler action can be identified as a complexified analog of free energy. Complexification follows both from the fact that g1/2 is real/imaginary in Euclidian/Minkowskian space-time regions. Complex values are also implied by the proposed identification of the values of Kähler coupling strength in terms of zeros and pole of Riemann zeta in turn identifiable as poles of the so called fermionic zeta defining number theoretic partition function for fermions (see this). The thermodynamical for Kähler action with volume term is Gibbs free energy G= F-TS= E-TS+PV playing key role in chemistry.

  4. The boundary conditions at the ends of space-time surfaces at boundaries of CD generalize appropriately and symmetries of WCW remain as such. At light-like boundaries between Minkowskian and Euclidian regions boundary conditions must be generalized. In Minkowkian regions volume can be very large but only the Euclidian regions contribute to Kähler function so that vacuum functional can be non-vanishing for arbitrarily large space-time surfaces since exponent of Minkowskian Kähler action is a phase factor.

  5. One can worry about almost topological QFT property. Although Kähler action from Minkowskian regions at least would reduce to Chern-Simons terms with rather general assumptions about preferred extremals, the extremely small cosmological term does not. Could one say that cosmological constant term is responsible for "almost"?

    It is interesting that the volume of manifold serves in algebraic geometry as topological invariant for hyperbolic manifolds, which look locally like hyperbolic spaces Hn=SO(n,1)/SO(n). See also the article Volumes of hyperbolic manifolds and mixed Tate motives. Now one would have n=4. It is probably too much to hope that space-time surfaces would be hyperbolic manifolds. In any case, by the extreme uniqueness of the preferred extremal property expressed by strong form of holography the volume of space-time surface could also now serve as topological invariant in some sense as I have earlier proposed. What is intriguing is that AdSn appearing in AdS/CFT correspondence is Lorentzian analogue Hn.

To sum up, the twistor lift of the dynamics of Kähler action allows to understand the origin of Planck length and cosmological constant. Here the earlier picture has been incomplete. Also the size scale of large voids and two fundamental biological length scales appear. p-Adic length scale hypothesis is realized in terms of the scaling factor of the 6-D Käler action defining giving rise to a dimensionless coupling constant. What is most remarkable that since only M4 and CP2 allow twistor space with Kähler structure, TGD is completely unique in twistor formulation.

Addition: I have modified the posting. I had a short one night-relationship with the attractive looking idea that the fiber of Minkowskian twistor space could be hyperbolic sphere H2 with Minkowskian signature of metric. I however realized my mistake and returned back to S2 from my nightly wanderings and did my best to delete all traces about what happened. I deeply regret my digression. I was also fluctuating between two alternatives for the signature of S2. Totally timelike (3 time-like dimensions altogether!) or totally space-like. The latter option turned out to be correct: I have been correct about un-physicality of 3 time-like dimensions. A further issue was whether Planck length is fundamental length or whether it emerges: it seems to be fundamental length.

See the the article From Principles to Diagrams or the chapter TGD Variant of Twistor Story of "Towards M-matrix".

For a summary of earlier postings see Links to the latest progress in TGD.

Saturday, February 06, 2016

Nematicity and high Tc superconductivity

Waterloo physicists discover new properties of superconductivity is the title of article popurazing the article of David Hawthorn, Canada Research Chair Michel Gingras, doctoral student Andrew Achkar and post-doctoral student Zhihao Hao published in Science.

There is a dose of hype involved. As a matter of fact, it has been known for years that electrons flow along stripes, kind of highways in high Tc superconductors: I know this quite well since I have proposed TGD inspired model explaining this (see this and this )!

The effect is known as nematicity and means that electron orbitals break lattice symmetries and align themselves like a series of rods. Nematicity in long length scales occurs a temperatures below the critical point for super-conductivity. In above mentioned cuprate CuO2 is studied. For non-optimal doping the critical temperature for transition to macroscopic superconductivity is below the maximal critical temperature. Long length scale nematicity is observed in these phases.

In second article it is however reported that nematicity is in fact preserved above critical temperature as a local order -at least up to the upper critical temperature, which is not easy to understand in the BCS theory of superconductivity. One can say that the stripes are short and short-lived so that genuine super-conductivity cannot take place.

These two observations yield further support for TGD inspired model of high Tc superconductivity and bio-superconductivity. It is known that antiferromagnetism is essential for the phase transition to superconductivity but Maxwellian view about electromagnetism and standard quantum theory do not make it easy to understand how. Magnetic flux tube is the first basic new notion provided by TGD. Flux tubes carry dark electrons with scaled up Planck constant heff =n×h: this is second new notion. This implies scaling up of quantal length scales and in this manner makes also super-conductivity possible.

Magnetic flux tubes in antiferromagnetic materials form short loops. At the upper critical point they however reconnect with some probability to form loops with look locally like parallel flux tubes carrying magnetic fields in opposite directions. The probability of reverse phase transition is so large than there is a competion. The members of Cooper pairs are at parallel flux tubes and have opposite spins so that the net spin of pair vanishes: S=0. At the first critical temperature the average length and lifetime of flux tube highways are too short for macroscopic super-conductivity. At lower critical temperature all flux tubes re-connect permantently average length of pathways becomes long enough.

This phase transition is mathematically analogous to percolation in which water seeping through sand layer wets it completely. The competion between the phases between these two temperatures corresponds to quantum criticality in which phase transitions heff/h=n1 ←→n2 take place in both directions (n1 =1 is the most plausible first guess). Earlier I did not fully realize that Zero Energy Ontology provides an elegant description for the situation (see this and this). The reason was that I though that quantum criticality occurs at single critical temperature rather than temperature interval. Nematicity is detected locally below upper critical temperature and in long length scales below lower critical temperature.

During last years it has become clear that condensed matter physicists are discovering with increasing pace the physics predicted by TGD . Same happens in biology. It is a pity that particle physicists have missed the train so badly. They are still trying to cook up something from super string models which have been dead for years. The first reason is essentially sociological: the fight for funding has led to what might be politely called "aggressive competion". Being the best is not enough and there is a temptation to use tricks, which prevent others showing publicly that they have something interesting to say. ArXiv censorship is excellent tool in this respect. Second problem is hopelessly narrow specialization and technicalization: colleague can be defined by telling the algorithms that he is applying. Colleagues do not see physics for particle physics - or even worse, for "physics" or superstrings and branes in 10,11, or 12 dimensions.

For a summary of earlier postings see Links to the latest progress in TGD.

Thursday, February 04, 2016

From Principles to Diagrams

The generalization of twistor diagrams to TGD framework has been very inspiring (and also frightening) mission impossible and allowed to gain deep insights about what TGD diagrams could be mathematically. I of course cannot provide explicit formulas but the general structure for the construction of twistorial amplitudes in N=4 SUSY suggests an analogous construction in TGD thanks to huge symmetries of TGD and unique twistorial properties of M4× CP2.

I try to summarize the big vision. Several guiding principles are involved and have gradually evolved to a coherent whole.

Imbedding space is twistorially unique

It took roughly 36 years to learn that M4 and CP2 are twistorially unique.

  1. Space-times are surfaces in M4× CP2. M4 and CP2 are unique 4-manifolds in the sense that both allow twistor space with Kähler structure: Kähler structure is the crucial concept. Strictly speaking, M4 and its Euclidian variant E4 allow both twistor space and the twistor space of M4 is Minkowskian variant T(M4)= SU(2,2)/SU(2,1)× U(1) of 6-D twistor space CP3= SU(4)/SU(3)× U(1) of E4. The twistor space of CP2 is 6-D T(CP2)= SU(3)/U(1)× U(1), the space for the choices of quantization axes of color hypercharge and isospin.

  2. This leads to a proposal for the formulation of TGD in which space-time surfaces X4 in H are lifted to twistor spaces X6, which are sphere bundles over X4 and such that they are surfaces in 12-D product space T(M4)× T(CP2) such the twistor structure of X4 are in some sense induced from that of T(M4)× T(CP2).
    What is nice in this formulation is that one can use all the machinery of algebraic geometry so powerful in superstring theory (Calabi-Yau manifolds). It was a complete surprise that a clear examination of this ideas leads to a profound understanding of the relationship between TGD and GRT (this will be discussed in later blog posting). Planck length emerges whereas fundamental constant as also cosmological constant emerges dynamically from the length scale parameter appearing in 6-D Kähler action. One can say, that twistor extension is absolutely essential for really understanding the gravitational interactions although the modification of Kähler action is extremely small due to
    the huge value of length scale defined by cosmological constant.

  3. Masslessness (masslessness in complex sense for virtual particles in twistorialization) is essential condition for twistorialization. In TGD massless is masslessness in 8-D sense for the representations of superconformal algebras. This suggests that 8-D variant of twistors makes sense. 8-dimensionality indeed allows octonionic structure in the tangent space of imbedding space. One can also define octonionic gamma matrices and this allows a possible generalization of 4-D twistors to 8-D ones using generalization of sigma matrices representing quaternionic units to to octonionic sigma "matrices" essential for the notion of twistors. These octonion units do not of course allow matrix representation unless one restricts to units in some quaternionic subspace of octonions. Space-time surfaces would be associative and thus have quaternionic tangent space at each point satisfying some additional conditions.

Strong form of holography

Strong form of holography (SH) following from general coordinate invariance (GCI) for space-times as surfaces states that the data assignable to string world sheets and partonic 2-surfaces allows to code for scattering amplitudes. The boundaries of string world sheets at the space-like 3-surfaces defining the ends of space-time surfaces at boundaries of causal diamonds (CDs) and the fermionic lines along light-like orbits of partonic 2-surfaces representing lines of generalized Feynman diagrams become the basic elements in the generalization of twistor diagrams (I will not use the attribute "Feynman" in precise sense, one could replace it with "twistor" or even drop away). One can assign fermionic lines massless in 8-D sense to flux tubes, which can also be braided.

One obtains a fractal hierarchy of braids with strands, which are braids themselves. At the lowest level one has braids for which fermionic lines are braided. This fractal hierarchy is unavoidable and means generalization of the ordinary Feynman diagram. I have considered some implications of this hierarchy (see this).

The existence of WCW demands maximal symmetries

Quantum TGD reduces to the construction of Kähler geometry of infinite-D "world of classical worlds" (WCW), of associated spinor structure, and of modes of WCW spinor fields which are purely classical entities and quantum jump remains the only genuinely quantal element of quantum TGD. Quantization without quantization, would Wheeler say.

By its infinite-dimensionality, the mere mathematical existence of the Kähler geometry of WCW requires maximal isometries. Physics is completely fixed by the mere condition that its mathematical description exists.

Super-symplectic and other symmetries of WCW are in decisive role. These symmetry algebras have conformal structure and generalize and extend the conformal symmetries of string models (Kac-Moody algebras in particular). These symmetries give also rise to the hierarchy of Planck constants. The super-symplectic symmetries extend to a Yangian algebra, whose generators are polylocal in the sense that they involve products of generators associated with different partonic surfaces. These symmetries leave scattering amplitudes invariant. This is an immensely powerful constraint, which remains to be understood.

Quantum criticality

Quantum criticality (QC) of TGD Universe is a further principle. QC implies that Kähler coupling strength is mathematically analogous to critical temperature and has a discrete spectrum. Coupling constant evolution is replaced with a discrete evolution as function of p-adic length scale: sequence of jumps from criticality to a more refined criticality or vice versa (in spin glass energy landscape you at bottom of well containing smaller wells and you go to the bottom of smaller well).

This implies that either all radiative corrections (loops) sum up to zero (QFT limit) or that diagrams containing loops correspond to the same scattering amplitude as tree diagrams so that loops can eliminated by transforming them to arbitrary small ones and snipping away moving the end points of internal lines along the lines of diagram (fundamental description).

Quantum criticality at the level of super-conformal symmetries leads to the hierarchy of Planck constants heff=n× h labelling a hierarchy of sub-algebras of super-symplectic and other conformal algebras isomorphic to the full algebra. Physical interpretation is in terms of dark matter hierarchy. One has conformal symmetry breaking without conformal symmetry breaking as Wheeler would put it.

Physics as generalized number theory, number theoretical universality

Physics as generalized number theory vision has important implications. Adelic physics is one of them. Adelic physics implied by number theoretic universality (NTU) requires that physics in real and various p-adic numbers fields and their extensions can be obtained from the physics in their intersection corresponding to an extension of rationals. This is also enormously powerful condition and the success of p-adic length scale hypothesis and p-adic mass calculations can be understood in the adelic context.

In TGD inspired theory of consciousness various p-adic physics serve as correlates of cognition and p-adic space-time sheets can be seen as cognitive representations, "thought bubbles". NTU is closely related to SH. String world sheets and partonic 2-surfaces with parameters (WCW coordinates) characterizing them in the intersection of rationals can be continued to space-time surfaces by preferred extremal property but not always. In p-adic context the fact that p-adic integration constants depend on finite number of pinary digits makes the continuation easy but in real context this need not be possible always. It is always possible to imagine something but not always actualize it!

Scattering diagrams as computations

Quantum criticality as possibility to eliminate loops has a number theoretic interpretation. Generalized Feynman diagram can be interpreted as a representation of a computation connecting given set X of algebraic objects to second set Y of them (initial and final states in scattering) (trivial example: X={3,4} → 3× 4 = 12 → 2× 6 → {2,6}=Y. The 3-vertices (a× b=c) and their time-reversals represent algebraic product and co-product.

There is a huge symmetry: all diagrams representing computation connecting given X and Y must produce the same amplitude and there must exist minimal computation. The task of finding this computation is like finding the simplest representation for the formula X=Y and the noble purpose of math teachers is that we should learn to find it during our school days. This generalizes the duality symmetry of old fashioned string models: one can transform any diagram to a tree diagram without loops. This corresponds to quantum criticality in TGD: coupling constants do not evolve. The evolution is actually there but discrete and corresponds to infinite number critical values for Kahler coupling strength analogous to temperature.

Reduction of diagrams with loops to braided tree-diagrams

  1. In TGD pointlike particles are replaced with 3-surfaces and by SH by partonic 2-surfaces. The important implication of 3-dimensionality is braiding. The fermionic lines inside light-like orbits of partonic 2-surfaces can be knotted and linked - that is braided (this is dynamical braiding analogous to dance). Also the fermionic strings connecting partonic 2-surfaces at space-like 3-surfaces at boundaries of causal diamonds (CDs) are braided (space-like braiding).

    Therefore ordinary Feynman diagrams are not enough and one must allow braiding for tree diagrams. One can also imagine of starting from braids and allowing 3-vertices for their strands (product and co-product above). It is difficult to imagine what this braiding could mean. It is better to imagine braid and allow the strands to fuse and split (annihilation and pair creation vertices).

  2. This braiding gives rise in the planar projection representation of braids to a generalization of non-planar Feynman diagrams. Non-planar diagrams are the basic unsolved problem of twistor approach and have prevented its development to a full theory allowing to construct exact expressions for the full scattering amplitudes (I remember however that Nima Arkani-Hamed et al have conjectured that non-planar amplitudes could be constructed by some procedure: they notice the role of permutation group and talk also about braidings (describable using covering groups of permutation groups)). In TGD framework the non-planar Feynman diagrams correspond to non-trivial braids for which the projection of braid to plane has crossing lines, say a and b, and one must decide whether the line a goes over b or vice versa.

  3. An interesting open question is whether one must sum over all braidings or whether one can choose only single braiding. Choice of single braiding might be possible and reflect the failure of string determinism for Kähler action and it would be favored by TGD as almost topological quantum field theory (TQFT) vision in which Kähler action for preferred extremal is topological invariant.

Scattering amplitudes as generalized braid invariants

The last big idea is the reduction of quantum TGD to generalized knot/braid theory (I have talked also about TGD as almost TQFT). The scattering amplitude can be identified as a generalized braid invariant and could be constructed by the generalization of the recursive procedure transforming in a step-by-step manner given braided tree diagram to a non-braided tree diagram: essentially what Alexander the Great did for Gordian knot but tying the pieces together after cutting. At each step one must express amplitude as superposition of amplitudes associated with the different outcomes of splitting followed by reconnection. This procedure transforms braided tree diagram to a non-braided tree diagrams and the outcome is the scattering amplitude!

See the the article From Principles to Diagrams or the chapter TGD Variant of Twistor Story of "Towards M-matrix".

For a summary of earlier postings see Links to the latest progress in TGD.

Wednesday, February 03, 2016

Quantal heat conduction in scale of one meter!

The finnish research group led by Mikko Möttönen working at Aalto University has made several highly interesting contributions to condensed matter physics during last years (see the popular articles about condensed matter magnetic monopoles and about tying quantum knots: both contributions are interesting also from TGD point of view). This morning I read about a new contribution published in Nature ).

What has been shown in the recent work is that quantal thermal conductivity is possible for wires of 1 meter when the heat is transferred by photons. This length is by a factor 104 longer than in the earlier experiments. The improvement is amazing and the popular article tells that it could mean a revolution in quantum computations since heat spoling the quantum coherence can be carried out very effectively and in controlled manner from the computer. Quantal thermal conductivity means that the transfer of energy along wire takes place without dissipation.

To understand what is involved consider first some basic definitions. Thermal conductivity k is defined by the formula j= k∇ T, where j is the energy current per unit area and T the temperature. In practice it is convenient to use thermal power obtained by integrating the heat current over the transversal area of the wire to get the heat current dQ/dt as analog of electric current I. The thermal conductance g for a wire allowing approximation as 1-D structure is given by conductivity divided by the length of the wire: the power transmitted is P= gΔ T, g=k/L.

One can deduce a formula for the conductance at the the limit when the wire is ballistic meaning that no dissipation occurs. For instance, superconducting wire is a good candidate for this kind of channel and is used in the measurement. The conductance at the limit of quantum limited heat conduction is an integer multiple of conductance quantum g0= kB2π2T/3h: g=ng0. Here the sum is over parallel channels. What is remarkable is quantization and independence on the length of the wire. Once the heat carriers are in wire, the heat is transferred since dissipation is not present.

A completely analogous formula holds true for electric conductance along ballistic wire: now g would be integer multiple of g0=σ/L= 2e2/h. Note that in 2-D system quantum Hall conductance (not conductivity) is integer (or more generally some rational) multiple of σ0= e2/h. The formula in the case of conductance can be "derived" heuristically from Uncertainty Principle Δ EΔ t=h plus putting Δ E = eΔ V as difference of Coulomb energy and Δ t= e/I=e L/ΔV=e/g0.

The essential prerequisite for quantal conduction is that the length of the wire is much shorter than the wavelength assignable to the carrier of heat or of thermal energy: λ>> L. It is interesting to find how well this condition is satisfied in the recent case. The wavelength of the photons involved with the transfer should be much longer than 1 meter. An order of magnitude for the energy of photons involve and thus for the frequency and wavelength can be deduced from the thermal energy of photons in the system. The electron temperatures considered are in the range of 10-100 mK roughly. Kelvin corresponds to 10-4 eV (this is more or less all that I learned in thermodynamics course in student days) and eV corresponds to 1.24 microns. This temperature range roughly corresponds to thermal energy range of 10-6-10-5 eV. The wave wavelength corresponding to maximal intensity of blackbody radiation is in the range of 2.3-23 centimeters. One can of course ask whether the condition λ >> L=1 m is consistent with this. A specialist would be needed to answer this question. Note that the gap energy .45 meV of superconductor defines energy scale for Josephson radiation generated by super-conductor: this energy would correspond to about 2 mm wavelength much below one 1 meter. This energy does not correspond to the energy scale of thermal photons.

I am of course unable to say anything interesting about the experiment itself but cannot avoid mentioning the hierarchy of Planck constants. If one has E= hefff, heff=n× h instead of E= hf, the condition λ>> L can be easily satisfied. For superconducting wire this would be true for superconducting magnetic flux tubes in TGD Universe and maybe it could be true also for photons, if they are dark and travel along them. One can even consider the possibility that quantal heat conductivity is possible over much longer wire lengths than 1 m. Showing this to be the case, would provide strong support for the hierarchy of Planck constants.

There are several interesting questions to be pondered in TGD framework. Could one identify classical space-time correlates for the quantization of conductance? Could one understand how classical thermodynamics differs from quantum thermodynamics? What quantum thermodynamics could actually mean? There are several rather obvious ideas.

  1. Space-time surfaces are preferred extremals of Kähler action satisfying extremely powerful conditions boiling down to strong form of holography stating that string world sheets and partonic 2-surfaces basically dictate the classical space-time dynamics. Fermions are localized to string world sheets from the condition that electromagnetic charge is well-defined for spinor modes (classical W fields must vanish at the support of spinor modes).

    This picture is blurred as one goes to GRT-standard model limit of TGD and space-time sheets are lumped together to form a region of Minkowski space with metric which deviates from Minkowski metric by the sum of the deviations of the induced metrics from Minkowski metric. Also gauge potentials are defined as sums of induced gauge potentials. Classical thermodynamics would naturally correspond to this limit. Obviously the extreme simplicity of single sheeted dynamics is lost.

  2. Magnetic flux tubes to which one can assign space-like fermionic strings connecting partonic 2-surfaces are excellent candidates for the space-time correlates of wires and at the fundamental level the 1-dimensionality of wires is exact notion at the level of fermions. The quantization of conductance would be universal phenomenon blurred by the GRT-QFT approximation.

    The conductance for single magnetic flux tube would be the conductance quantum determined by preferred extremal property, by the boundary conditions coded by the electric voltage for electric conduction and by the temperatures for heat conduction. The quantization of conductances could be understood in terms of preferred extremal property. m-multiple of conductance would correspond to m flux tubes defining parallel wires. One should check whether also fractional conductances coming as rational m/n are possible as in the case of fractional quantum Hall effect and assignable to the hierarchy of Planck constants heff=n × h as the proportionality of quantum of conductance to 1/h suggests.

  3. One can go even further and ask whether the notion of temperature could make sense at quantum level. Quantum TGD can be regarded formally as a "complex square root" of thermodynamics. Single particle wave functions in Zero Energy Ontology (ZEO) can be regarded formally as "complex square roots" of thermodynamical partition functions and the analog of thermodynamical ensemble is realized by modulus squared of single particle wave function.

    In particular, p-adic thermodynamics used for mass calculations can be replaced with its "complex square root" and the p-adic temperature associated with mass squared (rather than energy) is quantized and has spectrum Tp= log(p)/n using suitable unit for mass squared (see this).

    Whether also ordinary thermodynamical ensembles have square roots at single particle level (this would mean thermodynamical holography with members of ensemble representing ensemble!) is not clear. I have considered the possibility that cell membrane as generalized Josephson junction is describable using square root of thermodynamics (see this). In ZEO this would allow to describe as zero energy states transitions in which initial and final states of event corresponding to zero energy state have different temperatures.

    Square root of thermodynamics might also allow to make sense about the idea of entropic gravity, which as such is in conflict with experimental facts (see this).

See the article Quantization of thermal conductance and quantum thermodynamics or the chapter Criticality and dark matter.

For a summary of earlier postings see Links to the latest progress in TGD.

Tuesday, February 02, 2016

Is Planck length really fundamental length?

I have had intense discussions in a small group and I have been protesting against Planck length mystics assuming that there are objects with size of Planck length or discretization of space-time using Planck length/time as a unit. I noticed some sloppiness in my latest argument, made it more precise, and decided to attach it here.

  1. The notions of Planck time/length/mass are outcomes of dimensional analysis. You take, G, c and - this is important- Planck constant ℏ. For notational convenience I will choose the units so that I have c=1 in the following. You form a combination with dimensions of time/length and call it Planck time/length given by lP=( ℏ G)1/2. Since square root of ℏ appears, Planck length is an essentially quantal notion: one cannot have it in classical theory. This trivial observation could serve as a very important guideline for anyone dreaming of unified theory of all interactions!

    A couple of days ago I realized that could take also G, e2 with dimensions of ℏ, and c =1 and form constant with dimensions of time and this would, make also sense as a classical notion if one starts from electrodynamics or gauge theory although interpretation as genuine geometric size is far from obvious. It would be differ by (e2/ℏ)1/2 from Planck time being slightly smaller. This deduction of almost Planck length is however not possible if one takes fine structure constant as the fundamental dimensionless constant and identifies e2=α×4πℏc as derived quantal fundamental constant! The situation is very delicate!

    The overall important point is that you do not have Planck time without quantum theory unless you include electromagnetism or gauge interactions by assuming that e rather than fine structure constant is the fundamental constant. This has not been noticed. Probably the reason is that very few people really think how one ends up with the notion of Planck time. Young people want to amaze their professor with skilful calculations. Thinking is more difficult and does not fill hundreds of pages with impressive patterns of symbols.

  2. Planck length has no geometric interpretation unless you artificially assume it. To my opinion fundamental length must have a straightforward geometric interpretation already at classical level. In TGD it would be size scale of CP2. In string theory compactifications this kind of scales emerge but in completely ad hoc manner.
TGD does not predict Planck scale as classical fundamental length although it would first seem that in TGD framework Kähler coupling strength g2K could replace e2 replacing ℏ so that one would obtain something rather near to Planck length already in classical theory. This is not the case!

The point is that induced Kähler form is dimensionless unlike ordinary gauge field: one can think that electroweak U(1) gauge potential at QFT limit is obtained from the dimensionless Kähler gauge potential by multiplying it with 1/gK: AK,μ →AU(1),μ/gK. The inverse of this scaling is done routinely in path integral approach to gauge theories. At fundamental level one has only the dimensionless αK available and gK2= αK×4πℏc can emerge only as a derived quantity but only in quantum TGD!

G and Planck length/time/mass thus emerge from quantum TGD.

  1. Classically CP2 size scale R is the only quantity with dimensions of length in TGD. Can TGD predict G? G appears in the GRT-gauge theory limit of TGD, when space-time sheets as nearly parallel surfaces in M4× CP2 are lumped together to form with single one - the space-time of GRT note representable in generic case as a surface in M4×CP2.

  2. Can one predict the value of G? I have only a guess for the formula for G in terms of R, ℏ, and exponent of Kähler action for a deformation of what I call CP2 type vacuum extremal having interpretation as line of generalized Feynman diagram representing graviton. The value of this exponent for the line characterized the order of magnitude for the coupling strength assignable to the particle exchange characterized by the line. I believe that the general form of the formula is correct and conforms with the general form of vacuum functional but a lot remains to be understood.

  3. Planck mass, which also depends is proportional to square root of ℏ emerges in TGD as a dimensional parameter. hgr= heff characterizes magnetic flux tube connecting masses M and m and mediating graviton exchanges. The important point is that hgr/h= heff/h=n >1 is possible only if the product Mm is larger than Planck mass squared. Planck mass is a central parameter in quantum TGD but in a manner totally different from that in string models or loop quantum gravity. For instance, Planck mass as stringy mass scale is replaced in TGD with ℏ/R .

    The perturbative description of gravitational interactions with objects for which produce of masses is above Planck mass squared fails because the perturbation theory in powers of Gmm does not converge for Mm larger than Planck mass squared. Mother Nature has solved the problem and theoreticians need not worry about this: problem disappears in a phase transition changing h to heff/h=n=hgr/h= Gmm/hv0 and inducing also other nice things such as quantum coherence in even astrophysical scales essential for life. Mother Nature is generous. v0 has dimensions of velocity and perturbation expansion is now in powers of v0/c<1.

    This phase transition has interpretation in terms of fractal super-conformal symmetry breaking leading from superconformal algebra to a sub-algebra isomorphic with it. It has also interpretation at the level of space-time surfaces. Space-time sheets become singular n=heff/h-fold coverings. Also an interpretation in terms of quantum criticality is possible. A further interpretation is in terms of generation of dark matter as phases with non-standard value of Planck constant.

  4. Once G has successfully emerged, one can construct from G , ℏ and c=1 also a parameter with dimensions of diffusion constant D (this is trivial: dimensions of D are length squared over time: multiply Planck length by c (=1 now)) and it seems that this parameter is natural in TGD. It is not a geometric scale but describes deviation from classical theory and appears in the root mean squared for quantum fluctuations of the distance between the ends of light-like orbit of partonic 2-surface as function between distance between the ends (see this and this). One could of course say that it is D which emerges first and then comes GRT-gauge theory limit of TGD. This is somewhat a matter of taste.
So: one should be extremely careful with dimensional analysis when one applies it at the level of fundamental theories! String theory and loop quantum gravity have demonstrated this rather convincingly;-).

For a summary of earlier postings see Links to the latest progress in TGD.

Could cold fusion solve some problems of the standard view about nucleosynthesis?

The theory of nucleosynthesis involves several uncertainties and it is interesting to see whether interstellar cold fusion could provide mechanisms allowing improved understanding of the observed abundances. There are several problems: D abundance is too low unless one assumes the presence of dark matter/energy during Big Bang nucleosynthesis (BBN); there are two Lithium anomalies; there is evidence for the synthesis of boron during BBN; for large redshifts the observed metallic abundances are lower than predicted. The observed abundances of light nuclei are higher than predicted and require that so called cosmic ray spallation producing them via nuclear fission induced by cosmic rays. The understanding of abundances of nuclei heavier than Fe require supernova nucleosynthesis: the problem is that supernova 1987A did not provide support for the r-process.

The idea of dark cold fusion could be taken more seriously if it helped to improve the recent view about nucleosynthesis. In and additional section to the article Cold fusion again I try to develop a systematic view about how cold fusion could help in these problems. I take as a starting point the earlier model for cold dark fusion discussed in the above link and also in blog postings: see this, this, and this. This model could be seen as generalization of supernova nucleosynthesis in which dark variant of neutron and proton capture gives rise to more massive isotopes. Also a variant allowing the capture of dark alpha particle can be considered. Besides this pure standard physcis modification of Big Bang nucleosynthesis is proposed based on the resonant alpha capture of 7Li allowing to produce more Boron and perhaps explain second Li anomaly.

See the article Cold fusion again or the chapter Cold fusion again of "Hyper-finite factors and dark matter hierarchy".

For a summary of earlier postings see Links to the latest progress in TGD.

Saturday, January 30, 2016

Three atmospheric puzzles with a common solution

The motivation of this a posting came from a popular article telling about the theory of Earle Williams and colleagues explaining so called D-region ledge below 80 km. The mystery is that are no free electrons in lower atmosphere: one application is to radio communications. Radio waves propagate in ionosphere like in wave guide but not in lower atmosphere which is insulator.

I decided to look whether TGD inspired new physics might tell something interesting about the problem but the conclusion was negative. This search process ended as I realized that the model of Williams and colleagues might elegantly explain also two other poorly understood phenomena related to Earth's atmosphere.

  1. Consider first D-region ledge. The incoming solar UV radiation generates free electrons, mostly by ionization of nitric oxide NO by kicking away the unpaired electron. This happens in the entire atmosphere. The electron density however goes to practically zero at lower heights and during night-time D-region disappears entirely. As a consequence, atmosphere is rather poor conductor of electricity. The idea of the article is that the dust generated as small meteors burn in the atmosphere into small dust particles, which then bind the free electrons generated by UV radiation and then gradually fall down to ground.

  2. The stability of Earth's electric field is second mystery. Earth is negatively charged generating so called fair weather potential giving rise to electric field about 100-300 V/m at the surface of Earth and going to zero at heights about 1000 km. We do not understand the reason for why Earth is negatively charged.

    Even worse, a simple estimate using the value of electric field and estimate for ionic conductivity shows that it should take only a time of about 500 seconds for this negative charge to be lost by positive ionic currents from ionosphere (see this)! How Earth can preserve its negative charge? What mechanism prevents these currents from flowing or compensates them with opposite currents? Thunderstorms and electric clouds have been proposed as mechanisms bringing negative charge to Earth. But this only shifts the problem to that of understanding how electric clouds and thunder are generated.

    The model explaining D-region ledge could solve also this problem. There would be an ohmic positive ion current to earth and small ohmic electron current upwards in Earth's electric field. But besides this there would be downwards "gravitational" current of negatively charged dust particles compensating the ohmic current in equilibrium! Electrons could drift upwards to D-region from ground but could travel down as free travellers of dust particles!

    In purely plasma physical mood one would neglect gravitation altogether since the ratio mg/qE of gravitational and electric forces would be about 10-12 for electron. For dust particles with low enough ratio Q/m ratio one cannot neglect gravitation! Note that dust particles with critical Q/m ratio for which electric and gravitational forces compensate each other could remain stationary in atmosphere. The critical radio would be of order 10-9. From this one can estimate the critical charge of say water droplet, bacterium, or levitating meditator;-).

  3. The origin of electric clouds and thunder storms is a third mystery. In the regions with thunder clouds at heights about 10 km the electric field can become about 103 V/m and lightnings are generated when the strengths becomes large than that required by di-electric breakdown in air. Between thunder clouds and Earth the electric field usually changes its sign. We do not really understand how the large negative charge of thunder cloud and positive charge at the surface of Earth below it are generated.

    The model of D-region ledge suggests also a mechanism for the generation of thunder clouds. The electrically charged dust stucks cloud to like dust to water or ice so that the cloud becomes electrically charged. Since the dust does not reach ground and positive ionic current reaches it, the local electric field of Earth changes sign and eventually reaches the value needed for di-electric breakdown.

  4. Lightnings are found to have a strange feature: the energies of electrons can be relativistic and gamma rays are observed. This does not fit with the standard view. For this I have proposed explanation in terms of dark electrons travelling along magnetic flux tubes without dissipation and thus accelerating to to relativistic energies of order 105, one fifth of electron mass. Dark matter in TGD sense is indeed at criticality and the criticality corresponds now to the dielectric breakdown.
    My first thought was that this model suitably extended might throw light also to the above three mysteries but I soon realized that standard physics is enough: the magnetic flux tubes containing dark ions and electrons represent a small effect.

For a summary of earlier postings see Links to the latest progress in TGD.