We have rushed our way to this main point. The general framework that we have laid out up to this point is mathematically consistent, we will become more precise in discussing the physics behind these equations in the posts that follow. The reason why we have described the general framework is so that we could discuss the key equation
Σ_i(σ^t_i)(Â_p) – Σ_k(A_p) = Σ_l(σ^t_l)Ω_p
We define relationships between t_i, t_k and t_l in terms of;
t_i = t_l + 1
t_k + t_l = 2t_i
t_k = t_i + 1
Relations between particle numbers.
We noted earlier that we wanted to pay particular attention to the inverse relationship between N and P_1, or more generally P_i and t_i. If P_i is a measure of space, the distance between points, then as N approaches infinity we expect P_1 to approach zero, but to never actually be zero. In other words, as the distance between two points becomes zero N becomes infinitely large.
The reason for highlighting this relationship is that it hides an important symmetry which we will try to expose in future posts.
t’Hooft builds a Quantum Field Theory based on this step of the discretization process. He builds a wavefunction ¦Ψ_N>. In this next picture, we are generalising the idea of a divisor N.
We suggest that it is possible to define N as an element of a set of such numbers, each of which is defined by some observer. Each element in that set being a positive integer value. That is;
t_i = {m,r,t, … }
t_k = {N, S, V, …}
t_l = {j, z, w, …}
We have arbitrary groups of observers, of which N is an observation by a single observer. K is an invariant amongst all observers. We take its value to be the same in every frame. This is possible for quantities like the Planck length which is a constant. For example, we imagine that K = l_p = Planck length. P_i becomes some measure that depends on the values t_i and e_i so that it is considered an invariant quantity by all observers.
For example, suppose that t_i = t_1 for i = 1 and that t_1 = N. That is, at some position 1 in the set t_i we locate the quantity (N).
P_1 = K/t_1 + e_1
P_1 = K/N + e_1
Another way of thinking about this is to realise that N is inversely proportional to P_1. We can allow K to vary along with P_1 and hold e_1 as a constant of the theory.
For example, we can allow
P_1(t) = K(t)/t_1 + e_1
Which by differentation in the the time variable (t) we write;
dP_1(t)/dt = dK(t)dt (1/t_1) + de_1/dt
dP_1(t)/dt = dK(t)dt (1/t_1)
\dot{P _1} = \dot{K} (1/t_1)
P_1 is one of many theories possible. What we will discover is that it will be necessary for us to work in the general space
*dP_i = dK (1/t_i)
* There is an interesting physics associated with this p-form as we will see.
To connect with t’Hooft, we may tentatively suggest a function ¦Ψ_t_i>.
We begin our detour with a picture of handwritten notes. At this point in time it will not be entirely necessary to go deeply into Quantum Field Theory.
We only need to be aware that a crucial step in quantizing a field theory is to make spacetime discrete by dividing it into lattices. Perhaps I should be more careful here. Gerard t’Hooft is especially interested, page 15 “The Conceptual Basis of Quantum Field Theory”, in the discretization of time. He writes that; T⁄ Ν = δt.
Hello again WordPresserz!
We had started a discussion on inverse p-forms but sadly, immediate pressing concerns took us away from pursuing that rather intriguing line. However, it was not all in vain. I have managed to dig a little deeper and uncover exciting uses for the vector fields we delved into.
I wish to now take a detour onto a more fruitful exercise. I hope you will tag along – pun intended – as we explore some interesting uses of the vector field idea that kicked off this blog.
Hello,
Continuing with our theme; we can more explicitly write an equation
(1/χ) • (χ•d) = d
where 1/χ is the inverse to χ
and χ is a brilliant little symmetric (0,2) tensor that comes in useful if we imagine that somehow a and b define a 2-d manifold. It is a useful tool because we can use it to imagine that we can define a group based upon the multiplication rules of the new exterior derivative defined above.
By this I mean that…
(1/χ) • (χ•d) ≠ (χ•d)(1/χ)
This is part of the reason why we discussed the Lie algebra because that algebra allows us to define, in algebraic terms, the loss of symmetry shown above. Also, the Lie algebra opens up opportunity to describe group elements such as exp(χ•d).
It is by using group theoretic-ish methods that we will try to define the inverse of the p-form. By inverse of the p-form we mean to understand that
(χ•d)Φ = χ•dΦ = Φ
where dΦ is the p-form
The idea is to prove, using physics, that the above equation is correct. My own view is that a mathematical proof is not so far off.
Catchy title. But seriously, I came across this thing which behaves as an inverse to the p-form.
χ•d = a∂/∂a + (b/a)∂/∂b
where we allow χ = 1 + 1/a to be a 2×2 diagonal matrix. We allow this operator to act on a scalar Φ as…
(χ•d)(Φ) = a∂Φ/∂a + (b/a)∂Φ/∂b
(χ•d)(Φ) = χ•dΦ ≠ d•χΦ
This formula, I think, possibly opens up exciting possibilities. For example, we can imagine that there is some sort of anti-symmetry that can be captured by the Lie bracket so that
[χ•d , Φ] = (χ•d)(Φ) – (Φ)(χ•d)
It is interesting to note some of the properties, if at all, of this rendition of the Lie bracket such as its operation on another scalar field given by φ.
[χ•d , Φ] (φ) = (χ•dΦ)(φ) – ((Φ)(χ•d) )(φ)
[χ•d , Φ] φ = (χ•dΦ)φ – Φ(χ•dφ)
Perhaps the usefulness of the Lie bracket is that it measures the extent to which symmetry is lost, and is itself an operator on scalars. There are many interesting properties to this.
Hiya,
As I say up there, my purpose is to take physics to the next level. Of-course, that’s a goal…an objective, a dream perhaps, but a focus nevertheless.
I also say that quite confidently, don’t I? Like I know something…or something. Maybe I do, or maybe I’m just off my tool. Whatever.
Ok. So next physics…what’s that about? I mean,what’s it really? I describe it as the step above string theory – so if string theory is 21st century physics fallen in the 20th, then this is 22nd physics fallen in…nah forget that.
Just physics.
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