- 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.
-
- 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.
-
- 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.
-
- 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.
-
- 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.
-
- 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:
-
----------------- a + bj + cj + dk
-
----------------
where i, j, and k are the square roots of -1.
-
- 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.
-
----------------- (a)
pitch is the angle with the horizontal
-
----------------
(b) yaw is the turning of left or
right
-
---------------- ( 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.
-
- 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“.
-
- 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.
-
-
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.
-
- 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.
-
- 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.
-
-
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.
-
- 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.
-
- 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.
-
May 16, 2023 STRING
THEORY - Need 8-Dimensional 1284
4008
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--------------------- --- Tuesday, May 16, 2023 ---------------------------------
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