Comparing TGD- and QFT based descriptions of particle interactions

Marko Manninen made interesting questions related to the relationship between TGD and quantum field theories (QFTs). In the following, I will try to summarize an overview of this relationship in the recent view about quantum TGD. I have developed the latest view of quantum TGD in various articles (see this, this, this, this, this, and this).

Differences between QFT and quantum TGD

Several key ideas related to quantum TGD distinguish between TGD and QFTs.

1. The basic problem of QFT is that it involves only an algebraic description of particles. An explicit geometric and topological description is missing but is implicitly present since the algebraic structure of QFTs expresses the point-like character of the particles via commutation and anticommutation relations for the quantum fields assigned to the particles.

   In the string models, the point-like particle is replaced by a string, and in the string field theory, the quantum field Ψ(x) is replaced by the stringy quantum field Ψ(string), where the "string" corresponds a point in the infinite-D space of string configurations.

   In TGD, the quantum field Ψ(x) is replaced by a formally classical spinor field Ψ (Bohr orbit). The 4-D Bohr orbits are preferred extremals of classical action satisfying holography forced by general coordinate invariance without path integral and represent points of the "world of classical worlds" (WCW). The components of Ψ correspond to multi-fermion states, which are pairs of ordinary 3-D many-fermion states at the boundaries of causal diamond (CD).

   The gamma matrices of the WCW spinor structure are linear combinations of fermionic oscillator operators for the second quantized free spinor field of H. They anticommute to the WCW metric, which is uniquely determined by the maximal isometries for WCW guaranteeing the existence of the spinor connection. Physics is unique from its existence, as implied also by the twistor lift and number theoretic vision and of course, by the standard model symmetries and fields.

2. In TGD, the notion of a classical particle as a 3-surface moving along 4-D "Bohr orbit" as the counterpart of world-line and string world sheet is an exact aspect of quantum theory at the fundamental level. The notions of classical 3-space and particle are unified. This is not the case in QFT and the notion of a Bohr orbit does not exist in QFTs. TGD view of course conforms with the empirical reality: particle physics is much more than measuring of the correlation functions for quantum fields.

   Quantum TGD is a generalization of wave mechanics defined in the space of Bohr orbits. The Bohr orbit corresponds to holography realized as a generalized holomorphy generalizing 2-D complex structure to its 4-D counterpart, which I call Hamilton-Jacobi structures (see this). Classical physics becomes an exact part of quantum physics in the sense that Bohr orbits are solutions of classical field equations as analogs of complex 4-surfaces in complex M⁴×CP₂ defined as roots of two generalized complex functions. The space of these 4-D Bohr orbits gives the WCW (see this), which corresponds to the configuration space of an electron in ordinary wave mechanics.

3. The spinor fields of H are needed to define the spinor structure in WCW. The spinor fields of H are the free spinor fields in H coupling to its spinor connection of H. The Dirac equation can be solved exactly and second quantization is trivial.

   This determines the fermionic propagators in H and induces them at the space-time surfaces. The propagation of fermions is thus trivialized. All that remains is to identify the vertices. But there is also a problem: how to avoid the separate conservation of fermion and antifermion numbers. This will be discussed below.

4. At the fermion level, all elementary particles, including bosons, can be said to be made up of fermions and antifermions, which at the basic level correspond to light-like world lines on 3-D parton trajectories, which are the light-like 3-D interfaces of Minkowski spacetime sheets and the wormhole contacts connecting them.

   The light-like world lines of fermions are boundaries of 2-D string world sheets and they connect the 3-D light-like partonic orbits bounding different 4-D wormhole contacts to each other. The 2-D surfaces are analogues of the strings of the string models.

5. In TGD, classical boson fields are induced fields and no attempt is made to quantize them. Bosons as elementary particles are bound states of fermions and antifermions. This is extraordinarily elegant since the expressions of the induced gauge fields in terms of embedding space coordinates and their gradients are extremely non-linear as also the action principle. This makes standard quantization of classical boson fields using path integral or operator formalism a hopeless task.

   There is however a problem: how to describe the creation of a pair of fermions and, in a special case, the corresponding bosons, when there are no primary boson fields? Can one avoid the separate conservation of the fermion and the antifermion numbers?

Description of interactions in TGD

Many-particle interactions have two aspects: the classical geometric description, which QFTs do not allow, and the description in terms of fermions (bosons do not appear as primary quantum fields in TGD).

1. At the classical level, particle reactions correspond to topological reactions, where the 3-surface breaks, for example, into two. This is exactly what we see in particle experiments quite concretely. For instance, a closed monopole flux tube representing an elementary particle decomposes to two in a 3-particle vertex.

   There is field-particle duality realized geometrically. The minimal surface as a holomorphic solution of the field equations defines the generalization of the light-like world line of a massless particle as a Bohr orbit as a 4-surface. The equations of the minimal surface in turn state the vanishing of the generalized acceleration of a 3-D particle identified as 3-surface.

   At the field level, minimal surfaces satisfy the analogs of the field equations of a massless free field. They are valid everywhere except at 2-D singularities associated with 3-D light-like parton trajectories. At singularities the minimal surface equation fails since the generalized acceleration becomes infinite rather than vanishing. The analog of the Brownian particle experiences acceleration: there is an "edge" on the track.

   At singularities, the field equations of the whole action are valid, but are not separately true for various parts of the action. Generalized holomorphy breaks down. These 2-D singularities are completely analogous to the poles of an analytic function in 2-D case and there is analogy with the 2-D electrostatics, where the poles of analytic function correspond to point charges and cuts to line charges.

   This gives the TGD counterparts of Einstein's equations, analogs of geodesic equations, and also the analogy Newton's F=ma. Everything interesting is localized at 2-D singularities defining the vertices. The generalized 8-D acceleration H^k defined by the trace of the second fundamental form, is localized on these 2-D parton surfaces, vertices. One has a generalization of Brownian motion for a particle-like object defined by a partonic 2-surface or equivalently for a particle as 3-surface. Intriguingly, Brownian motion has been known for a century and Einstein wrote his first paper after his thesis about Brownian motion!

   Singularities correspond to sources of fermion fields and are associated with various conserved fermion currents: just like in QFTs. For a given spacetime surface, the source- vertex - is a discrete set of 2-D partonic surface just as charges correspond to poles of analytic function in 2-D electrostatics.

   At the classical level, the 2-D singularities of the minimal surfaces therefore correspond to vertices and are localized to the light-like paths of parton surfaces where the generalized holomorphy breaks down and the generalized acceleration H^k is there non-vanishing and infinite.

Description of the interaction vertices

1. How to get the TGD counterparts of the QFT vertices?

   Vertices typically contain a fermion and an antifermion and the gauge potential, which is second quantized. Now, classical gauge potentials are not second quantized. How to obtain the basic gauge theory vertices?

   This is where the standard approximation of QFTs helps intuition: replace the quantized boson field with a classical one. This gives the vertex corresponding to the creation of a pair of fermions. Thanks to that, only the fermion and the sum of the antifermion numbers are conserved and the theory does not reduce to a free field theory. One should be able to do the same now. However, the precise formulation of this vision is far from trivial.

2. The modified Dirac action should give elementary particle vertices for a given Bohr trajectory.

   There are two options:

   1. Modified gammas are defined as contractions of ordinary gamma matrices of H with the canonical momentum currents associated with the classical action defining the space-time surface. Supersymmetry is now exact: besides color and Poincare super generators there is an infinite number of conserved super symplectic generators and infinitesimal generalized superholomorphisms.

      This option does not work: the modified Dirac equation implies that the Dirac action and also vertices vanish identically. Although one has partonic 2-surfaces as singularities of minimal surfaces defining vertices, the theory is trivial because the usual perturbation theory does not work.

   2. Modified gamma matrices are replaced by the induced gamma matrices defined by the volume term (cosmological term of the classical action). Supersymmetry is broken but only at the 2-D vertices. The anticommutator of the induced gammas gives the induced metric. This is not true for the modified gammas defined by the entire action: in this case the anticommutators are rather complex, being bilinear in the canonical momentum currents. Is it possible to have a non-trivial theory despite the breaking of supersymmetry at vertices or or does the supersymmetry breaking make possible a non-trivial theory? This seems to be the case.
      1. In 2-D vertices, the generalized acceleration field H^k is proportional to the 2-D delta function and gives rise to the graviton and Higgs vertices. One obtains also the vertices related to gauge bosons from the coupling of the induced spinor field to induced spinor connection. Only the couplings to electroweak gauge potentials and U(1) K&aum;hler gauge potential of M⁴ are obtained. The failure of the generalized holomorphy is absolutely essential.
      2. Color degrees of freedom are completely analogous to translational degrees of freedom since color quantum numbers are not spin-like in TGD. Strong interactions are vectorial and correspond to Kähler gauge potentials.
      3. Generalized Brownian motion gives the vertices. One obtains the equivalents of Einstein's and Newton's equations at the vertices. The M⁴ part M^k of the generalized acceleration is related to the gravitons and the CP₂ part S^k to the Higgs field. Spin J=2 for graviton is due to the rotational motion of the closed monopole flux tube associated with the gravitation giving an additional unit of spin besides the spin of H^k, which is S=1.
   3. Consider now the description of fermion pair creation.
      1. Intuitively, the creation of a fermion pair (and thus also a boson) corresponds to the fermion turning backwards in time. At the level of the geometry of the space-time surface, this corresponds to the partonic 2-surface turning backwards in time, and the same happens to the corresponding fermion line. Turning back in time means that effectively the fermion current is not conserved: if one does not take into account that the parton surface turns in the other direction of time, the fermion disappears effectively and the current must has a singular divergence. This is what the divergence of the generalized acceleration means.
      2. This implies that the separate conservation is lost for fermion and antifermion numbers. This means breaking of supersymmetry, of masslessness, of generalized holomorphy and also the generation of the analog of Higgs vacuum excitation as CP₂ part S^k of the generalized acceleration H^k. The Higgs vacuum expectation is only at the vertices. But this is exactly what is actually wanted! No separate symmetry breaking mechanism is needed!
      3. The failure of the generalized holomorphy at the 2-D vertex means that the holomorphic partonic orbit turns at the singularity to an antiholomorphic one. For the annihilation vertex it could occur only for the hypercomplex part of the generalized complex structure.
      4. Remarkably, the states associated with connected 4-surfaces consist of either fermions or antifermions but not both. This explains matter antimatter asymmetry if quantum coherence is possible in arbitrarily long scales. In TGD, space-time surfaces decompose to regions containing either matter or antimatter and, by the presence of quantum coherence even in cosmological scales, these regions can be very large. The quantum coherence in large scales is implied by the number theoretic vision predicting a hierarchy of Planck constants labelling phases of ordinary matter behaving like dark matter (see for instance this).
      5. What is the precise mathematical formulation of this vision? This is where a completely unique feature of 4-dimensional manifolds comes in: they allow exotic smooth structures. Exotic smooth structure is the standard smooth structure with lower-dimensional defects. In TGD, the defects correspond in TGD to 2-D parton vertices as "edges" of Brownian motion. In the exotic smooth structure, the edge disappears and everything is soft. Pair creation and non-trivial theory is possible only in dimension D=4 (see this and this ).

      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.