Tuesday, May 16, 2023

4008 - STRING THEORY MATH - Need 8-Dimensional ?

 

-    4008  -    STRING  THEORY  MATH  -  Need 8-Dimensional?   This review discusses how math was invented to solve problems encountered in nature.  3 dimensions in space and 1 dimension in time is familiar.  But, in fundamental particle physics math is used with 8 dimensional numbers.  Maybe 10 dimensions explains reality?  That is what String Theory is exploring.


--------------  4008   -  STRING  THEORY  MATH  -  Need 8-Dimensional ?

-    The “Real Number System” is a line of numbers starting with zero and increasing in the positive direction to the right, and, increasing in negative numbers to the left.  The points in between the integer numbers are the fractions.  The line is one-dimensional because a specific point requires only one Real Number to specify it (-1).

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-     “Complex Numbers” are of the form “a + bi', where “a” and “b” are Real Numbers and “i” is the Imaginary Number equal to the square root of -1.  Using Complex Numbers we can describe points on a plane, not just on a line.  The Imaginary Numbers form the vertical axis of this plane and the Real Numbers form the horizontal axis on this plane.  The axes are perpendicular and intersect at zero.

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-    A special way to look at this plane of real numbers and imaginary numbers is to look at addition, subtraction, multiplication, and division as geometric manipulations in the plane.

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-    For the real numbers adding or subtracting slides the Real Axis line to the left or to the right.  Multiplying or dividing by any positive number stretches or squashes the line.  For example:  Dividing by 2 squashes the line moving all the  points twice as close to each other.  Multiplying by -1 flips the line over, rotating it by 180 degrees.

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-    The same type of geometric manipulations happen with complex numbers

( a + bi).  Adding i to  (2 + i)  =  2 + 2i which slides the plane up by 1 and to the right by 2.  The plane moves left or right by “a” amount and up or down but the “b” amount.  Multiplying stretches or squashes the plane but also rotates it.

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 -    Multiplying by “i” rotates the plane a quarter turn.  Multiplying by ”i” twice rotates the plane a full half-turn.  Division has the effect of shrinking the plane and rotating it in the opposite direction.

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-    The Complex Number ( a + bi) can have a short hand notation of simply (a , b).  In this notation adding 2 is simply ( a + 2, b+ 2).  Multiplication is ( i * 2i )  =          ( -2 ) .  Division is (  2i/2i )  =  1 .  Multiplication rotates counterclockwise by 90 degrees.  Division rotates clockwise by 90 degrees.

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-     It was later learned the multiplication and division done in 2 dimensions could not be done in 3 dimensions. To get the math to work the dimensions had to be 1,2,4, or 8 dimensions.  In the 4 dimension case the math is:

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-----------------  a + bj + cj + dk

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----------------  where i, j, and k are the square roots of -1.

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-    We need points in 4-dimensional numbers in order to describe changes in 3-dimensional space.  The three numbers describe the rotations of  pitch, yaw and roll.

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-----------------  (a)   pitch is the angle with the horizontal

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----------------  (b)  yaw is the turning of left or right

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---------------- ( c )    roll is the angle of the planes wings as it makes the maneuver

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----------------  (d)  The 4th number again describes the stretching and shrinking of the plane.

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-    The math rules do change when dealing with 4-dimensional numbers.  The multiplication of numbers in 1 or 2 dimensions “ commute”.  That is it  makes no difference in which order you multiply  2 * 3  =  6  The same as 3 * 2 = 6.  However, with 4-dimensional numbers the order of multiplication matters.  4-dimensional numbers are  “non-commutative“.

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-  The non-commutative phenomenon is usually explained using the rotation of a book.  Book in hand, flip it vertically, rotate it and you are facing the edges of the pages.  Again, book in hand,   rotate it, flip it vertically and the spine of the book is facing you.  The order of rotation matters, rotation does not commute.

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-  8-dimension numbers present even more math exceptions.  Not only are these numbers Non-commutative, they are Non-Associative  The Associative law is ( x*y) *z  =  x*(y*z).   1-dimensional subtraction is Non-Associative.  (3 - 2) - 1  = 0  This is not the same as 3 - (2 - 1)  =  2.  In 1 and 2 dimensions multiplications and rotations are associative but non-commutative.

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-    The 8-dimensional math has today found its way into String Theory and Particle Physics.  Every fundamental particle of matter has its partner particle that is the force carrier.   

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-     The Electron has the force carrier, the Photon.  The Quarks have the force carrier, the Gluon.  The concept of Super symmetry requires that the laws of physics remain unchanged if we exchange all the Matter particles with all the Force Carrier particles.  They are perfectly symmetrical particles.

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-     Today, in the field of Quantum Mechanics the math used describes the Matter particles as waves and their motion is described in wave equations.  The math used to describe the wave motion of the force particles is vectors.   

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-     If physicists use one-dimensional math to describe a string, and they add a second dimension for time the string traces out a plane over time.  Adding time gives us another dimension.  So, instead of the math working for 2, 4, and 8 dimensions it now works for 3, 4, 6, and 10 dimensions.  Only the 10-dimensional math appears to work without some problem anomalies.  10 dimensional string theory uses 8 dimensional math.

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-    So far String Theory and 8-dimensional numbers have produced beautiful mathematics.  We just wish we could prove it worked to describe something in nature’s reality.  So far the math is an invention and not a discovery.  However, it will not be the first time that the math came first and physicists discovered how reality worked later. 

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May 16, 2023   STRING  THEORY  -  Need 8-Dimensional   1284       4008                                                                                                                         

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