# Overlooked wavefunction duality

1. Jul 21, 2003

### Loren Booda

An interchange between variable action and Planck's constant in
conventional wavefunctions represents a spectrum of virtual states
that invert standard eigennumber solutions. The resultant "inverse
wavefunctions" predict discrete values for virtual actions (in general
those less than or equal to 2h). The inverse (virtual) wavefunction
obeys both the de Broglie and Einstein postulates, and can be
expressed in a linear Schroedinger equation. Conventional (real) and
virtual wavefunctions may interfere to generate the family of
subatomic particles from associated superstring vibration and winding
numbers. The virtual wavefunction endows phase space with information
nested locally (=<2h), yet prerequisite for seeming non-local
correlations. That real and virtual wavefunctions evolve from a
common phase space origin helps to explain this anomaly.

I simply ask you, especially those familiar with quantum field theory,
whether the virtual wavefunction could be a valid concept. More
details describing this overlooked modification to quantum mechanics
can be found in the first article of my website at
http://www.quantumdream.net

2. Jul 22, 2003

### Raavin

I'm an idiot but in lay terms are you talking about all wavefunctions (particles and their properties) being linked regardless of apparent distance and these wavefunctions interacting in a phase space where, if we are to use an analogue to 3d space, everything is colocated??

Raavin [?]

3. Jul 22, 2003

### ranyart

For a photon, it has a mirror anti-photon, it has a dual wavefunction, it is always entangled with its mirror particle.

The photon can exchange its momentum with its own anti-photons position, and vica versa, this is what entangled states are.

An ordinary observed(detected) photon is one that has un-tangled in the process of HUP needed to observe it.

The appearence of a Virtual wavefunction cannot manifest locally, in the near vicinity of real photons, it has to have no interference with its corresponding mirror partner, thus the space it occupies is the for the briefest moment, at the farthest location from its coupled counterpart, one could say that when a real photon is detected, its mirror self particle, is at the other end of the wavefunction, with no local interference registered.

4. Jul 23, 2003

### Loren Booda

Raavin, take spacetime evolving from the big bang as an analogy. The horizon problem states that for expansion following Hubble's law, the opposite horizons relative to the observer cannot now interchange information, but once could.

Likewise, I postulate mutually entangled phase space events to have arisen from a primordial shared state, and retain weak correlation. Specific real wavefunction actions expand outward from a value h/2 (h being Planck's constant), and corresponding virtual wavefunction actions progress inwardly from 2h. Translated to spacetime, an observation may virtually encompass great distances or particle velocities simultaneously.

The mapping of real action eigenvalues is reciprocally conformal to those virtual. The two, "polar" wavefunctions exhibit spherical symmetry (colocate) around a spherical shell of action h in phase space. They also create particles where there are mixed (real and virtual) states interfering.

5. Jul 23, 2003

### jeff

Was there something specific you came across that inspired you're idea?

6. Jul 24, 2003

### Loren Booda

jeff, that "The inverse (virtual) wavefunction obeys both the de Broglie and Einstein postulates, and can be expressed further in a linear Schroedinger equation." Also, the only justifications for a mathematical wavefunction entity are essentially identical for both "real" and "virtual" wavefunctions. For some applications in field theory, I believe, the virtual wavefunction is more efficient than its real counterpart.

7. Jul 24, 2003

### jeff

What are the de Broglie and Einstein postulates?

8. Jul 24, 2003

### Loren Booda

[lamb]=h/p and [nu]=E/h

9. Jul 24, 2003

### jeff

The second relation has always been attributed to Planck since he was the first to write it down.

To avoid confusion and arguments about semantics, would you mind laying out in detailed mathematical terms the precise correspondence between T-duality and "P-duality"?

10. Jul 24, 2003

### Loren Booda

For Planck, [nu]=E/h was more a mathematical fix, whereas Einstein applied it specifically to the physical particle nature of the quantum, directly measurable.

My P-duality, or phase duality, compares the solutions of wavefunctions with standard and inverted phase arguments. By phase arguments I mean those ratios therein between variable action (rp, say) and constant action (Planck's constant h, say), or vice versa. Again, examples are given beginning the first article of my website http://www.quantumdream.net.

These phase arguments, for the wavefunctions given, generate eigenvalues to an arbitrarily zero-valued eigenvalue equation 0=Im(exp(2[pi](rp/h))) or 0=Im(exp(2[pi](h/rp))). The solutions, in units of h, for the former conventional "real" wavefunction are the N/2, and for the latter inverse "virtual" wavefunction the 2/N-, where N and N- are nonzero integers.

Vibration and winding numbers of T-duality are given by analogous N and N-. Combining these components (see the chart in Brian Greene's popular book on superstrings, or page 29 of April 1996 Physics Today) is achieved equivalently by solving interfered P-dualistic wavefunctions.

11. Jul 25, 2003

### jeff

Loren,

Working in (1+1)-dimensional spacetime, apply p-duality to the three simplest of quantum mechanical systems: the free particle, the particle in a box, and the harmonic oscillator. Write down all observables along with their commutation relations and in particular their hamiltonians and associated spectra.

Also, discuss how classical physics is recovered from these quantum mechanical p-dual systems.

Last edited: Jul 25, 2003
12. Jul 26, 2003

### Loren Booda

Loren Booda

Free Particle with potential: conventional-real <==> inverse-virtual (')

Wavefunctions (Y):

Y=A(exp(2[pi]ixpx/h)) <==> Y'=A'(exp(2[pi]ih/x'(px')'))

Operators (*):

(px)*=-i(h/2[pi])(d/dx) <==> (1/(px')')*=(-i/2[pi]h)(d/d(1/x'))

(x*)K(x-x)=(x)K(x-x) <==> ((1/x')*)K(1/x'-1/x')=(1/x')K(1/x'-1/x') where K is the Kronecker delta function

Commutators:

[x*,p*]=ih/2[pi] <==> [(1/x')*,(1/(px')')*]=i/2[pi]h

Hamiltonians:

H*=(p*)2/2m+V(x)=-(h/2[pi])2/2m+V(x) <==> [See the 1+1 dimensional (time dependent) virtual Schroedinger's equation on page 4 of my website]

Spectra:

E=nh[nu] <==> E'=h[nu]'/n'

1/n <==> n'

[nu]=1/[tau] <==> [nu]'=1/[tau]'

Last edited: Jul 26, 2003
13. Jul 26, 2003

### jeff

14. Jul 26, 2003

### Loren Booda

Jeff, I am still working out bugs in the formulation of the Hamiltonians.

15. Jul 27, 2003

### Loren Booda

Hamiltonians:

H*=(p*)2/2m+V(x)=-((h/2[pi])2/2m)d2/dx2+V(x) <==>

(1/H')*=((1/p')*)22m+1/V(1/x')=-((2[pi]h)-22m)d2/d(x')2+1/V(x')

16. Jul 27, 2003

### jeff

Free particles by definition don't interact with any potential, so really the hamiltonian for a free particle consists of only the kinetic term, for which...

is the plane wave solution for a free particle of definite momentum p. However,

are the energy levels of a bound system, the quantum harmonic oscillator - though it's missing the ground state contribution h&nu;/2. The solutions to it's schrodinger equation are the hermite polynomials, not the plane wave you've given above.

In fact, only bound systems - as opposed to free ones - have quantized energy spectra.

The classical limit in any quantum theory must always be recovered by taking h &rarr; 0, in which case we see from the usual commutation relation - you've written it on the left - that the operators X and P commute as any operators must in this limit. However, precisely the opposite happens in the commutation relation you've written on the right which indicates that it doesn't make sense.

The bosonic string spectrum m2 = n2/R2+w2R2/&alpha;&prime;2+2(NL+NR-2)/&alpha;&prime; is invariant under R &rarr; &alpha;&prime;/R , n &harr; w. The reason that this "T-duality" symmetry is possible is that closed strings in addition to having momenta n like point particles, also have winding number w. This kind of symmetry related to spatial degrees of freedom is therefore not possible with ordinary point particles, and in fact what you're proposing, namely,

is not a symmetry because the "P-dual" spectrum differs from the conventional one.

Last edited: Jul 27, 2003
17. Jul 27, 2003

### Loren Booda

Thanks, jeff, you've given me a lot of quality information to think about, the best I've received to date concerning my P-duality idea.

Last edited: Jul 27, 2003