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Feynman checkerboard with two paths contributing to the sum for the propagator
from (x / εc, t / ε) = (0, 0) to (3, 7).


The Feynman Checkerboard or Relativistic Chessboard model was Richard Feynman’s sum-over-paths formulation of the kernel for a free spin ½ particle moving in one spatial dimension. It provides a representation of solutions of the Dirac equation in (1+1)-dimensional spacetime as discrete sums.

It can be puzzling as to why a rotation of 720 degrees or two turns is necessary to return to the original state. This comes about because in quantum theory the state of a particle or system is represented by a complex probability amplitude and then when a measurement is made on the system the probability of it coming out some way is given by the square of absolute value of the appropriate amplitude.

Say you send a particle into a system with a detector that can be rotated where the probabilities of it detecting some state are affected by the rotation. When the system is rotated through 360 degrees the observed output and physics are the same as at the start but the amplitudes are changed for a spin ½ particle by a factor of -1 or a phase shift of half of 360 degrees. When the probabilities are calculated the -1 is squared and equals a factor of one so the predicted physics is same as in the starting position. Also in a spin 1/2 particle there are only two spin stated and the amplitudes for both change by the same -1 factor so the interference effects are identical unlike the case for higher spins. The complex probability amplitudes are something of a theoretical construct and cannot be directly observed.

If the probability amplitudes changed by the same amount as the rotation of the equipment then they would have changed by a factor of -1 when the equipment was rotated by 180 degrees which when squared would predict the same output as at the start but this is wrong experimentally. If you rotate the detector 180 degrees the output with spin ½ particles can be different to what it would be if you did not hence the factor of a half is necessary to make the predictions of the theory match reality.

The Checkerboard model is important because it connects aspects of spin and chirality with propagation in spacetime and is the only sum-over-path formulation in which quantum phase is discrete at the level of the paths, taking only values corresponding to the 4th roots of unity.

Paul Davies's book The Goldilocks Enigma (2006) reviews the current state of the fine tuning debate in detail, and concludes by enumerating the following responses to that debate:

1. The absurd universe

Our universe just happens to be the way it is.

2. The unique universe

There is a deep underlying unity in physics which necessitates the universe being the way it is. Some Theory of Everything will explain why the various features of the Universe must have exactly the values that we see.

3. The multiverse

Multiple Universes exist, having all possible combinations of characteristics, and we inevitably find ourselves within a Universe that allows us to exist.

4. Creationism

A creator designed the Universe with the purpose of supporting complexity and the emergence of Intelligence.

5. The life principle

There is an underlying principle that constrains the universe to evolve towards life and mind.

6. The self-explaining universe

A closed explanatory or causal loop: "perhaps only universes with a capacity for consciousness can exist." This is Wheeler's Participatory Anthropic Principle (PAP).

7. The fake universe

We live inside a virtual reality simulation.

Hello darkness, my old friend
I've come to talk with you again
Because a vision softly creeping
Left its seeds while I was sleeping
And the vision that was planted in my brain
Still remains
Within the sound of silence

In restless dreams I walked alone
Narrow streets of cobblestone
'Neath the halo of a street lamp
I turn my collar to the cold and damp
When my eyes were stabbed by the flash of a neon light
That split the night
And touched the sound of silence

And in the naked light I saw
Ten thousand people maybe more
People talking without speaking
People hearing without listening
People writing songs that voices never shared
No one dared
Disturb the sound of silence

"Fools," said I, "you do not know
Silence like a cancer grows
Hear my words that I might teach you
Take my arms that I might reach you"
But my words like silent raindrops fell
And echoed in the wells of silence

And the people bowed and prayed
To the neon god they made
And the sign flashed out its warning
In the words that it was forming
And the sign said "The words of the prophets are written on the subway walls
And tenement halls
And whispered in the sound of silence

“Physics is like sex. Sure, it may give some practical
results, but that's not why we do it.” -R.P. Feynman

vs

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Δ=2,4*pow(q1,1/3)+pow(q2,1/3)

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