A more fundamental notation may be that of the prime factors of any number. This stems from the fact that any number which is in the state divisible by 4 must also lead to a number divisible by 2 if one is to disregard the strange non-existant numbers which appeared in the simple divisor notation. This means the states now correspond to the prime numbers \(2,3,5,7,11,13...\) where one is now omitted! The states will be bosonic and follow bosonic rules, this is because numbers such as \(9\) are divisible by \(3\), a prime, in two ways. This then creates a number line of states in the progression \[1 = |0,0,0,0,0,...> \\ 2 = |1,0,0,0,0,...> \\ 3 = |0,1,0,0,0,...> \\ 4 = |2,0,0,0,0,...> \\ 5 = |0,0,1,0,0,...> \\ 6 = |1,1,0,0,0,...> \\ 7 = |0,0,0,1,0,...> \\ 8 = |3,0,0,0,0,...> \\ 9 = |0,2,0,0,0,...> \\ 10= |1,0,1,0,0,...> \\ etc.\]
The benifits of the negative notation is the allowance of integer fractions for example \(\frac{2}{3}=|1,-1,0,..>\).
This representation is nice. A key feature is that the product of two numbers is the term for term addition of their two states! For example \(2\cdot3=6\) is \(|1,0,...>+|0,1,...>=|1,1,...>\). This is from the normal rules of number multiplication. We also then know that each state is unique to a single number. A division process is then the subtraction of two states. For example \(8/2=4\) is expressed \(|3,0,..>-|1,0,..>=|2,0,..>\). This implies that negative entries are allowed in the state vectors! Something quantum field theory does not normally have, therefore the coefficients on any operation must be made to reflect this, OR, this may be expressed in the condition that \(n/0 = \infty\) for certain operations. The concept of zero is again hard to define! \(0/n=0\), so we require a state that with anything taken from it is still the same state! This would lead to the state \[0=|\infty,\infty,\infty,...>\] curious! This also fits the definition that \(0\cdot n=0\).This state simply states that zero can be divided by any prime number any number of times, which is true. Then the concept of an infinity state could also be defined, requiring \(\infty \cdot n=\infty\) for any countable \(n\), and \(n / \infty = 0\) which leads to \[\infty=|-\infty,-\infty,-\infty,...>\] So infinity can be divided (negatively?) as many times as necessary.
Then roughly speaking \[\infty \cdot 0 = |-\infty,-\infty,..>+|\infty,\infty,..> =?|0,0,0,...>=1\]
And also \[0/0 = |\infty,\infty,..>-|\infty,\infty,..>=|0,0,..> = 1\]
However, these kind of things are not quite the point of this notation. Although if that is indeed the definition of zero... then it would be true that \[\prod_{p \in \mathbb{P}} p^{\infty}= \prod_{p \in \mathbb{P}} \prod_{k=0}^{\infty}p= 0\]
A strange result! This would also indicate that \[\prod_{p \in \mathbb{P}} p^{-\infty} = \prod_{p \in \mathbb{P}} \prod_{k=1}^{\infty} \frac{1}{p} = \infty\]
The useful property of these results in that integer fractions can be expressed as states \(|1,-1,0,..>=\frac{2}{3}\) for example.
The operation of addition for states is not simple!
Now to include bosonic style operators. Counting could be performed from 1 as \[a^+_1|0,0>=|1,0> \\ a^+_2a_1|1,0>=|0,1> \\ a^+_1a^+_1a_2|0,1>=|2,0> \\ a^+_3a_1a_1|2,0>=|0,0,1> \\ a^+_2a^+_1a_3|0,0,1>=|1,1>\] Where the operators required appear to have little pattern. An attempt to find a pattern is made. The pattern below is to take the last operator, apply a reversal of it’s terms, take the conjugate of each term, add a new operation to the front and repeat. The sequence of new operators to apply would need to be found.
Process to explain each operator (experimental) \[a^+_1 \to a^+_1 \to a_1 \to (a^+_2)a_1 \\ a^+_2a_1 \to a_1a^+_2 \to a^+_1a_2 \to (a^+_1)a^+_1a_2 \\ a^+_1a^+_1a_2 \to a_2a^+_1a^+_1 \to a^+_2a_1a_1 \to (a^+_3a_2)a^+_2a_1a_1 \\ a^+_3a_1a_1 \to a_1a_1a^+_3 \to a^+_1a^+_1a_3 \to (a^+_2a_1)a^+_1a^+_1a_3 \\ a^+_2a^+_1a_3 \to a_3a^+_1a^+_2 \to a^+_3a_1a_2 \to (a^+_4a_3)a^+_3a_1a_2 \\\]
There could be odd states such that \[|\infty,0,\infty,0,...>+|0,\infty,0,\infty,...>=|\infty,\infty,\infty,\infty,...>=0 \\ M \cdot N = 0 \\ M \cdot M = M \\ N \cdot N = N\]
Then for a ternary operation, repeating phaseshifted sequences like \[|\infty,0,0,\infty,...>+|0,\infty,0,0,...>+|0,0,\infty,0,...>=|\infty,\infty,\infty,...>=0 \\ A \cdot B \cdot C = 0 \\ A \cdot A = A \\ B \cdot B = B \\ C \cdot C = C \\ A \cdot B \\ B \cdot C \\ A \cdot C\]
So these structures can form objects like squareroots and cube roots etc. of 1. Similar to matrices, alternating patterns and such...
So what does the inner product of two divisor states mean? \[|0,0,...> \cdot |a,b,...> = 0a +0b + ... \\ |1,0,...> \cdot |1,1,0,..> =1 \\ |p_i> \cdot |p_j> = \delta^i_j \\ |\infty,\infty,..> \cdot |0,0,..> = \infty ?? \\ |\infty,\infty,..> \cdot |a,b,c,..> = \infty\]
Like state vecotors in a Hilbert space (probably are), the number \(1\) is orthoganol to everything. Always a zero state, even to itself! Any prime \(p_i\) is orthogonal to any other prime except itself. Any number state that shares a divisor is not orthogonal to that number!
Can see the zero state is possibly not orthogonal to the 1 state and is definately not orthogonal to any other state.
But then there are states such as \[|1,1,0> \cdot |1,1,1> = 2\] which are ’doubly’ non-orthogonal. They share two prime divisors. We can try to imagine a small number space. There is a vertical axis \(1\) which is orthogonal to a multi dimensional “plane” or volume below. In this multidimensional volume we could imagine a \(2\) axis and as \(3\) axis both orthogonal to each other and \(1\) so forming a right handed set. Then we accept a dimensional folding such that the \(0\) axis is also aligned with both \(2\) and \(3\). However, an axis for \(6\) would also be aligned with both \(2\) and \(3\). If we included no other numbers here, so a numeral system where only the values \(0,1,2,3\) and \(6\) exist, then \(6\) lies in the same orientation as \(0\)!! (Unless \(0\) is not orthogonal to \(1\)). If we repeat this exercise, by making the largest number which is the product of all defined axes, then it will be aligned with all of the numbers, except \(1\). Zero would also be aligned with all of the numbers, meaning the highest number created by the multiplication of all axis defined numbers is in the same orientation as the zero axis! Extrapolating this, infinity would be in the same orientation as zero.