Wavelets and Wigner Phase Space Density

Notes by Jack Sarfatti

 

The Wigner phase space density  of a “signal”  is

 

(1.1)

 

(1.2)

 

(1.2) is a generalized Parseval conservation relation of basis independence. What is the signal  in (1.1)?  I am now not interested in generality but in my macro-quantum vacuum model specifically. You can consider the raw signal as the bound pair quantum wave function of a single virtual electron-positron pair with coordinates in 4-dim globally flat spacetime of the “false micro-quantum vacuum” of completely incoherent unbound virtual electron-positron pairs. This is a kind of quasi-Yilmaz space that Hal Puthoff likes, but there is no collective emergent gravity in this false vacuum. Emergent gravity in Andre Sakharov’s sense out of the globally flat gauge force and source pre-geometrodynamic micro-quantum foam requires a Bose-Einstein condensation, or macroscopic occupation of the single-virtual pair bound state  where  is the center of mass coordinate and  is the relative coordinate between the virtual electron and the virtual positron.  That is, let

 

(1.3)

 

The pair density matrix is

 

(1.4)

 

The reduced pair density matrix in the center of mass coordinates is

 

(1.5)

 

Bose-Einstein condensation (“ODLRO”[1]) in the center of mass coordinates is

 

(1.6)

 

The first term on RHS of (1.6) is the smooth coherent nonrandom macro-quantum “superfluid”. The second term is the residual locally random micro-quantum normal fluid from Heisenberg uncertainty noise that in vacuum we associate with “zero point fluctuations”.

(1.7)

 is the “large macroscopic eigenvalue” mean number of virtual pairs in the same center of mass “mother wavelet”. The local macro-quantum BEC order parameter, whose phase modulation gives Einstein’s , is a coherent superposition of different integer occupation numbers of the mother wavelet. The density matrix for the center of mass motion of the pair in (1.6) is a correlation function. Define

 

(1.8)

 

The superfluid part of the Wigner phase space density, in special relativity notation, which is OK for pre-gravity micro-quantum gauge force and source “false vacuum”, is therefore,

(1.9)

 

Similarly, for the normal random noisy “zero point fluctuations”,

 

(1.10)

 

 

I now drop the cm subscript in the coordinate notation. Take as the basic (“mother”) wavelet, the effective center of mass wave packet  of a single virtual electron-positron pair. Define the wavelet basis

 

(1.11)

 

Define the self-observing macro-quantum vacuum integral wavelet transform as

 

(1.12)

where  is the Kleinert world crystal lattice spacing ~ Planck scale.

 

Shall we take a Wigner phase space density of a wavelet, or a wavelet transform of a Wigner phase space density? Both may be interesting. Let’s first look at the latter. Start with (1.9)’s . What about                    

(1.13)

 

The self-measuring wavelet transform of the superfluid Wigner phase space density is then

(1.14)

 

The adaptive phase space window area is

 

(1.15)

 

Where obviously

(1.16)