Electroweak Symmetry Breaking and supersymplectic symmetries

One of the hardest challenges in the development of the TGD based view of weak symmetry breaking was the fact that classical field equations allow space-time surfaces with finite but arbitrarily large size. For a fixed space-time surface, the induced gauge fields, including classical weak fields, are long ranged. On the other hand, the large mass for weak bosons would require a short correlation length. How can one understand this together with the fact that a photon has a long correlation length?

In zero energy ontology quantum states are superpositions of space-time surfaces as analogs of almost unique Bohr orbits of particles identified as 3-D surfaces. For some reason the superposition should be such that the quantum averages of weak gauge boson fields vanish below the weak scale whereas the quantum average of electromagnetic fields is non-vanishing.

This is indeed the case.

1. The supersymplectic symmetries form isometries of the world of classical worlds (WCW) and they act in CP₂ degrees of freedom as symplectic transformations leaving the CP₂ symplectic form J invariant and therefore also its contribution to the electromagnetic field since this part is the same for all space-time surfaces in the superposition of space-time surfaces as a representation of supersymplectic isometry group (as a special case a representation of color group).
2. In TGD, color and electroweak symmetries acting as holonomies are not independent and for the SU(2)_L part of induced spinor connection the symplectic transformations induce SU(2)_L × U(1)_R gauge transformation. This suggests that the quantum expectations of the induced weak fields over the space-time surfaces vanish above the quantum coherence scale. The averages of W and of the left-handed part of Z⁰ should therefore vanish.
3. ⟨Z⁰⟩ should vanish. For U(1)_R part of Z⁰, the action of gauge transformation is trivial in gauge theory. Now, however, the space-time surface changes under symplectic transformations and this could make the average of the right-handed part of Z⁰ vanishing. The vanishing of the average of the axial part of the Z⁰ is suggested by the partially conserved axial current hypothesis.

One can formulate this picture quantitatively.

1. The electromagnetic field [1][2] contains, besides the induced Kähler form, also the induced curvature form R₁₂, which couples vectorially. The conserved vector current hypothesis suggests that the average of R₁₂ is non-vanishing. One can express the neutral part of the induced gauge field in terms of induced spinor curvature and Kähler form J as:

   R₀₃ = 2(2e⁰ ∧ e³ + e¹ ∧ e²) = J + 2e⁰ ∧ e³,
   J = 2(e⁰ ∧ e³ + e¹ ∧ e²),
   R₁₂ = 2(e⁰ ∧ e³ + 2e¹ ∧ e²) = 3J - 2e⁰ ∧ e³.

2. The induced fields γ and Z⁰ (photon and Z-boson) can be expressed as:

   γ = 3J - sin²θ_W R₁₂,
   Z⁰ = 2R₀₃ = 2(J + 2e⁰ ∧ e³).

   The condition ⟨Z⁰⟩ = 0 gives 2⟨e⁰ ∧ e³⟩ = -2J, and this in turn gives ⟨R₁₂⟩ = 4J. The average over γ would be:

   ⟨γ⟩ = (3 - 4sin²θ_W)J.

   For sin²θ_W = 3/4, ⟨γ⟩ would vanish.

The quantum averages of classical weak fields quite generally vanish. What about correlation functions?

1. One expects that the correlators of classical weak fields as color invariants, and perhaps even symplectic invariants, are non-vanishing below the Compton length since in this kind of situation the points in the correlation function belong to the same 3-surface representing particle, such as hadron.
2. The intuitive picture is that in longer length scales one has disjoint 3-surfaces with a size scale of Compton length. If the states associated with two disjoint 3-surfaces are separately color invariant there are no correlations in color degrees of freedom and correlators reduce to the products of expectations of classical weak fields and vanish. This could also hold when the 3-surfaces are connected by flux tube bonds.

   Below the Compton length weak bosons would thus behave as correlated massless fields. The Compton lengths of weak bosons are proportional to the value of effective Planck constant h_eff and in living systems the Compton lengths are proposed to be even of the order of cell size. This would explain the mysterious chiral selection in living systems requiring large parity violation.

3. What about the averages and correlators of color gauge fields? Classical color gauge fields are proportional to the products of Hamiltonians of color isometries induced Kähler form and the expectations of color Hamiltonians give vanishing average above Compton length and therefore vanishing average. Correlators are non-vanishing below the hadron scale. Gluons do not propagate in long scales for the same reason as weak bosons. This is implied by color confinement, which has also classical description in the sense that 3-surfaces have necessarily a finite size.

   A large value of h_eff allows colored states even in biological scales below the Compton length since in this kind of situation the points in the correlation function belong to the same 3-surface representing particle, such as dark hadron.

See the article Reduction of standard model structure to CP₂ geometry and other key ideas of TGD or the chapter Appendix.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.