## Saturday, March 13, 2021

### Zero energy states as scattering amplitudes and subjective time evolution as sequence of SSFRs

Zero energy states code for the ordinary time evolution in the QFT sense described by the S-matrix. Construction of zero energy is reasonably well understood (see this, this, and this ).

This is not yet the whole story. One should also understand the subjective time evolution defined by a sequence of "small" state function reductions (SSFRs) as analogs of "weak" measurements followed now and then by BSFRs. How does the subjective time evolution fit with the QFT picture in which single particle zero energy states are planewaves associated with a fixed CD?

1. The size of CD increases at least in statistical sense during the sequence of SSFRs. This increase cannot correspond to M4 time translation in the sense of QFTs. Single unitary step followed by SSFR can be identified asa scaling of CD leaving the passive boundary of the CD invariant. One can assume a formation of an intermediate state which is quantum superposition over different size scales of CD: SSFR means localization selecting single size for CD. The subjective time evolution would correspond to a sequence of scalings of CD.
2. The view about subjective time evolution conforms with the picture of string models in which the Lorentz invariant scaling generator L0 takes the role of Hamiltonian identifiable in terms of mass squared operator allowing to overcome the problems with Poincare invariance. This view about subjective time evolution also conforms with super-symplectic and Kac-Moody symmetries of TGD.

One could perhaps say that the Minkowski time T as distance between the tips of CDs corresponds to exponentiated scaling: T= exp(L0t). If t has constant ticks, the ticks of T increase exponentially.

The precise dynamics of the unitary time evolutions preceding SSFRs has remained open.
1. The intuitive picture that the scalings of CDs gradually reveal the entire 4-surface determined by polynomial P in M8: the roots of P as "very special moments in the life of self" would correspond to the values of time coordinate for which SSFRs occur as one new root emerges. These moments as roots of the polynomialdefining the space-time surface would correspond to scalings of the size of both half-cones for which the space-time surfaces are mirror images. Only the upper half-cone would be dynamical in the sense that mental images as sub-CDs appear at "geometric now" and drift to the geometric future.
2. The scaling for the size of CD does not affect the momenta associated with fermions at the points of cognitive representation in X4⊂ M8 so that the scaling is not a genuine scaling of M4 coordinates which does not commute with momenta. Also the fact that L0 for super symplectic representations corresponds to mass squared operator means that it commutes with Poincare algebra so that M4 scaling cannot be in question.
3. The Hamiltonian defining the time evolution preceding SSFR could correspond to an exponentiation of the sum of the generators L0 for super-symplectic and super-Kac Moody representations and the parameter t in exponential corresponds to the scaling of CD assignable to the replaced of root rn with root rn+1 as value of M4 linear time (or energy in M8). L0 has a natural representation at light cone boundaries of CD as scalings of light-like radial coordinate.
4. Does the unitary evolution create a superposition over all over all scalings of CD and does SSFR measure the scale parameter and select just a single CD?

Ordoes the time evolution correspond to scaling? Is it perhaps determined by the increase of CD from the size determinedby the root rn as "geometric now" to the root rn+1 so that one would have a complete analogy with Hamiltonian evolution? The scaling would be the ratio rn+1/rn which is an algebraic number.

Hamiltonian time evolution is certainly the simplest option and predicts a fixed arrow of time during SSFR sequence. L0 identifiable essentially as a mass squared operator acts like conjugate for the logarithm of the logarithm of light-cone proper time for a given half-cone.

One can assume that L0 as the sum of generators associated with upper and lower half-cones if the fixed state at the lower half-cone is eigenstate of L0 not affect in time evolution by SSFRs.

How does this picture relate to p-adic thermodynamics in which thermodynamics is determined by partition function which would in real sector be regarded as a vacuum expectation value of an exponential exp(iL0t) of a Hamiltonian for imaginary time t=iβ β=1/T defined by temperature? Here L0 is proportional to mass squared operator.
1. In p-adic thermodynamics temperature T is dimensionless parameter and β=1/T is integer valued. The partition function as exponential exp(-H/T) is replaced with pβ L0), β=n, which has the desired behavior if L0 has integer spectrum. The exponential form eL0/TR), βR= nlog(p) equivalent in the real sector does not make sense p-adically since the p-adic exponential function has p-adic norm 1 if it exists p-adically.
2. The time evolution operator exp(-iL0t) for SSFRs (t would be the scaling parameter) makes sense for the extensions of p-adic numbers if the phase factors for eigenstates are roots of unity belonging to the extension. t= 2π k/n since L0 has integer spectrum. SSFRs would define a clock. The scalingexp(t)= exp(2π k/n) is however not consistent with the scaling by rn-1/rn.

Both the temperature and scaling parameter for time evolution by SSFRs would be quantized by number theoretical universality. p-Adic thermodynamics could have its origins in the subjective time evolution by SSFRs.

3. In the standard thermodynamics it is possible to unify temperature and time by introducing a complex time variable \tau = t+iβ, where β=1/T is inverse temperature. For the space-time surface in complexified M8, M4 time is complex and the real projection defines the 4-surface mapped to H. Could thermodynamics correspond to the imaginary part of the time coordinate?

Could one unify thermodynamics and quantum theory as I have indeed proposed: this proposal states that quantum TGD can be seen as a "complex square root" of thermodynamics. The exponentials U=exp(\tau L0/2) would define this complex square root and thermo-dynamical partition function would be given by UU= exp(-β L0).

See the article Is M8-H duality consistent with Fourier analysis at the level of M4× CP2?.

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