CHAPTER 6: THE BELL'S INEQUALITY

      As we know this year's nobel prize for physics was shared by three scientists for disproving Einstein's theory of relativity that the speed of light is the fastest in space and nothing can travel faster than light. How did they proved it practically? What mathematics did they used ? 

      This started in the year 1935 when the three physicists: Boris Podolsky, Nathan Rosen and Albert Einstein gave one Paradox named EPR Paradox. This paradox states that there exists some hidden variable which is responsible for the changing states between two quantum particles which are seperated at very large distance. In other words for e.g. take two balls one of black color and other of red color put them in two identical boxes and you don't know which color ball is in which box. Take away one box to another planet and keep one box to yourself and open the box as soon as you observe the color of one ball you will immediately come to know the color of the ball in another box. Here exists a local hidden variable which is the box through which we came to know the information of the balls and obviously there is no travelling of information. Also, there is a difference between local hidden variable and hidden variable, we will come across this difference later. Also, we can take another example of twins who look exactly same. They look identical. The local hidden variable here is genes which make them different. As soon as we detect genes we will come to know the names of the two twins. These examples was according to EPR Paradox in which everything is already fixed in nature and we can't change it. Einstein gave this thought experiment a paradox because in quantum mechanics information travels at a speed greater than speed of light which is not possible.

        But things don't work like this in quantum mechanics. Any hidden variables doesn't exist in quantum theory. In the above the color of the box is fixed i.e. it does not change inside the box. But in quantum theory the quantum particles exists in two states in both the boxes at the same time. As soon as we observe the spin of one of the particle, the opposite spin will appear on the other particle. Initially, both the particles exists in superposition state, and after observing it one of the state appears. Einstein did not agreed with the quantum theory at all. This was like playing a dice in which we don't know which number will appear until the dice movement stops and we observe that number. 

        Quantum theory more clearly can be proved by the double-slit experiment. In this experiment one one can take a long rectangular shape cardboard paper, make two slits in it horizonal direction. Place one screen behind the cardboard to detect the electron particles emitted by the laser. There is no fix path of the particles some electrons will travel through slit (S1) other will go through slit (S2). The pattern of the particles will be random on the screen. But here happens one weird thing. As soon as we keep the camera or try to observe, we will see the wave pattern on the screen. This is only possible when one particle travels through both the slits at the same time! This experiment proved that electron exists both in waveform and in particle form. 

        There were two groups of scientists one who believed quantum theory and who supported Einstein. Then, in 1965 one scientist named John Steward Bell gave mathematical expressions related to the hidden variable which is known as the bell's inequality. The derivation of the bell's inequality is shown in the image inserted below:

                                         

                                                        

 In equation 3 the multiplication |ai bi| is always +1 because as mentioned in the image the value of ai and bi is either +1 or -1 and the absolute value of the multiplication is always +1. Also the modulus is removed in the right side of equation 3 because if we take the modulus the value of the equation on the right side will always be zero. Equation 4 is the bells inequality which is correct in terms of algebra. And this inequality is for the case in which the hidden variable exists because in this case we gave each fixed values (+1 or -1) at one time due to which the above equation holds good. But in quantum theory one list can have 1 and -1 at the same time due to which the left hand side equation will be greater than the right hand side. For e.g. if we take the value of b and c +1 in the right hand side and on the left side if we take b=+1 and c=-1 then right hand side will be smaller than left hand side.

        Later on three scientists disproved the bell's inequality as mentioned above. They experimentally disproved the inequality and made a ground breaking research in quantum physics due to which they received the nobel prize  in physics. The theoretical and mathematical explanation of their experimental result will be given the next chapter.