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Fahad Shiftra (born ) is a Pakistani self-taught mathematical physicist. The notable achievements in his career have included studies into the role of the Higgs boson in the grand unification theory and the moment of inertia of objects moving in a Bose Einstein condensate. At present, the movement of particles through supercritical mediums such as a Bose Einstein condensate is not well understood, it is hoped his equation will reveal the secret of how a sub-atomic particle behaves when acting through these conditions.

## Personal life

Born in Karachi, Fahad Shiftra grew up in one of the poorest regions of Pakistan. Unable to go to University, Shiftra worked as a shoe cleaner until the age of 12 in order to support his family. However, by the age of 15, and with a family now able to care for itself, he began to purchase math and science textbooks from local scholars. In fact, it was one of the scholars that he bought the textbook from who recognised his potential as a mathematician. Prof. Rana Khalid Naeem of Karachi University wrote in 1996 that[1]

"I have rarely seen such an enthusiastic attitude towards mathematics. Fahads curiosity is a gift that could bring him great knowledge"

This prediction seemed to come true in 2000 when he presented his first paper to the physics professors and lecturers at New Delhi University. In his presentation Shiftra successfully displayed an understanding and ingenious ability to combine existing problems in the physics world in order to explain the behaviour of sub-atomic particles when under extreme conditions. One lecturer claimed that it was a miracle that a man without a university education behind him could understand and explain in such detail the complex world of quantum mechanics. At present, Shiftra is currently living with his wife Sasha in Karachi, Pakistan with their two sons Henry and Lennon.

## Work and theories

When studying the behaviour of particles under intense pressure, Shiftra noticed a relationship between a Fourier series wave and its movement in a Bose-Einstein condensate.

Taking the fundamental form of the particles chirality to be

$\left[ \frac{-{\hbar}^2}{2m}\frac{{\partial}^2}{\partial x^2} + V(\mathbf{x}) \right]|\psi(t)\rang = i \hbar \frac{\partial}{\partial t} |\psi(t)\rang,$

$|\phi_1 \cdots \phi_N \rang = \sqrt{\frac{\prod_j N_j!}{N!}} \sum_{p\in S_N} |\phi_{p(1)}\rang \cdots |\phi_{p(N)} \rang,$

He noticed that this bore a resemblance to the condition

$\sum_{n=1}^\infty\frac{\mu(n)}{n^s}$

For the Riemann hypothesis.

Combining this with the Fourier series for the wave, you arrive at

$D_\mu D^\mu \phi = -(\partial_t - ie A_0)^2 \phi + (\partial_i - ie A_i)^2 \phi = m^2 \phi\,$

However, this is not correct for an object moving in a Bose-Einstein condensate. So, integrating with limits z = 1 and y = ?1 and developing from previous equations you arrive at [2]

$S= \int_x (\partial_\mu \phi^* + ie A_\mu \phi^* ) (\partial_\nu \phi - ie A_\nu\phi)\eta^{\mu\nu} = \int_x |D \phi|^2\,$

Then, when combining this with the approximate relative Fourier series and its assocaited partial derivatives, you arrive at

$\frac{ \partial^{i+j+k} f}{ \partial x^i\, \partial y^j\, \partial z^k } = f^{(i, j, k)}.$

and

\begin{align}a_n &{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)\,dx = 0, \quad n \ge 0. \\b_n &{}= \frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)\, dx = -\frac{2}{n}\cos(n\pi) = 2 \, \frac{(-1)^{n+1}}{n}, \quad n \ge 1.\end{align}

Combined gives [3]:

\begin{align}\iint\limits_S \mathbf{F} \cdot \mathbf{n} \, dS &= \iiint\limits_W\left(\nabla\cdot\mathbf{F}\right) \, dV\\ &= 2\iiint\limits_W\left(1+y+z\right) \, dV\\ &= 2\iiint\limits_W \,dV + 2\iiint\limits_W y \,dV + 2\iiint\limits_W z \,dV.\end{align}

This gives a useful formula that is not yet entirely correct in a resistant material (such as a Bose-Einstein condensate)

Shiftra adapted the Dirac equation for an effective "packet" of particles travelling through the supercritical fluid to get

$J_0 = z_{\mathrm{S}} d_{\mathrm{F}} D_{\mathrm{F}} \int_{-\infty}^{0} \mathrm{exp}(\epsilon /d_{\mathrm{F}}) \; \mathrm{d} \epsilon \; = \; z_{\mathrm{S}} {d_{\mathrm{F}}}^2 D_{\mathrm{F}} \; = \; Z_{\mathrm{F}} D_{\mathrm{F}}, ..........$

where $Z_{\mathrm{F}} \; [=z_{\mathrm{S}} {d_{\mathrm{F}}}^2]$ is the effective supply for state F, and is defined by this equation. Strictly, the lower limit of the integral should be –KF, where KF is the Fermi Energy; but if dF is very much less than KF (which is always the case for a metal) then no significant contribution to the integral comes from energies below KF, and it can formally be extended to –8.

This is modelled by a Fermi-Dirac distribution in the form

$\bar{n}_i = \frac{1}{e^{(\epsilon_i-\mu) / k T} + 1}$

taking into account the effect of quantum tunnelling that would occur, which obeys the law

$\Phi'(x) = A(x) + i B(x) \,$
$A'(x) + A(x)^2 - B(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right)$

We would arrive at a simplified version of the law derived by Shiftra from these expressions

In order to make the correct changes to the formula, several dimensionless quantities and constants need to be introduced. These are, Planck's constant 'h', viscosity 'mu', Mitul velocity 'MitB' - A dimensionless ratio of velocities through and outside the condensate, Reynolds number 'Re', Nusselt number 'Nu', Amos number 'Ja', Lewis resonance 'Le' and it's associated inductance in Henry's 'H'.

With these, and the correct manipulation of the partial derivatives, you arrive at the following equation:

$\int\limits_{x_1=x_a}^{x_1=x_b} \ldots \int\limits_{x_n=x_a}^{x_n=x_b}\ \exp \left(\frac{{\rm MitB}}{\hbar}\int\limits_{t_Le}^{t_Ja} \mathcal Le(n_e(n_p),o(n_o), Re)\,\mathrm{d}Ld\right)\, \mathrm{d}x_n \cdots \mathrm{d}x_{1}$

Which is known as "Fahads law" [4]. This is a completely theoretical formula that can only be used when dealing with particles at high energy levels.

## References

1. ^ Karachi University Archives
2. ^ http://www.iap.uni-bonn.de/ag_weitz/publikationen/Weitz.pdf
3. ^ Super critical movement of point charges in a Bose-Einstein condensate-Fahad Shiftra, Prof. Asad Abidi and Dr. Samar Mubarakmand
4. ^ Derivation of a mathematical relationship between the relative movement of point charges and their associated viscosic medium - Dr. Samar Mubarakmand, Fahad Shiftra and Prof. Ian Nugent

Karachi, Meteorological Department of Pakistan.

Clark, John, O.E. (2004). The Essential Dictionary of Science. Barnes & Noble Books.

Potential Higgs Boson discovery: Higgs Boson: Glimpses of the God particle

### Cited references

1. Karachi University Physics department lecture archives.

2. Wolfram Mathworld Fourier Series developments.