So far, I have decided to make a base for the conjecture, a supposed set of mathematical axioms to demonstrate a connection between a virtual particle and a real particle. Once I have done that, I need to apply that for two virtual particle pairs connected to every real particle in the vacuum. Then I need to apply that to what I am currently trying to get my head round, the pauli matrices.
In other words you're going to just make stuff up or lift it from elsewhere and hope people swallow it.
There will be two systems as subsystems $$(S_1,S_2)$$ in the complex and real valued range, and it's description will rely on the state vector acting on the Hilber Space $$H$$.
State vectors don't act on Hilbert spaces, they are elements
in a Hilbert space. Well done, you managed 1 sentence before demonstrating you're a liar.
The representation in superspace has for these subsystems $$S_1,S_2|\psi> \in (N: \mathbb{C}, \mathbb{R})$$ so that the representation of the physical states of each subsystem are eingenfunctions of an operator - so we will have an observable excpectancy $$<\psi|A_{1,2}|\psi>$$ associated to $$S_1$$ and $$S_2$$.
What has superspace got to do with it, superspace involves extending space-time to include a set of Grassman variables, such a process is done in
supersymmetry, which Dirac's work has nothing to do with. You've clearly copied this from somewhere else and failed to understand the things it talks about are irrelevant to your claims.
This means in their respective phase spaces, they will be reflected through probability to have each subsystem reflect at least two possible eigenstate conditions $$\phi$$ and $$\phi*$$.
This isn't even a coherent sentence. And you haven't given an eigenstate 'condition', you've just given two expressions which are conjugate to one another.
The eignstates of the mixed-system represented by the supserposition of $$S_1+S_2$$ would then be given as:
$$\alpha_0 \chi \phi* \mathbb{V} \alpha \chi \phi$$
Systems don't have eigenstates, operators do. You are demonstrating you don't understand basic terminology. Again.
where $$V$$ denotes the ''or''
The what?
We therefore have a joint state simply as $$\chi_0,\phi*$$ and $$\chi,\phi$$
Spurious use of commas.
Now suppose on the Hilbert Space, we arrange it so that the range of $$c_n$$ is the one which follows a non-linear distribution through time:
$$|\Psi>=\sum_{n=1}^{\infty} \psi_n c_n$$
Your equation doesn't follow from your statement, which itself doesn't make sense. All you've done is say "I will write a state as the sum of a basis" which is something you can do in any Hilbert space, its why they are used. And you haven't given any time dependence, non-linear or not. The equation you give is a linear combination!
If the range of the $$n^{th}$$ power is effected by a noisy thermal background (or some kind of strong coupling on a system)
Power? You haven't given any expression involving powers. You have given an expression involving a countable basis which is indexed by an integer but that isn't a power. This is even more basic than quantum mechanics, its basic vectors!
on the system $$\mathbb{S}=(S_1,S_2)$$ of subsystems, their interaction terms $$U$$ can be given as:
$$U\psi_0= \alpha_0 \chi_0 \phi*+\alpha \chi \phi$$
You've again failed to understand the thing you've copying. You haven't given an interaction term, as might be given in a Lagrangian, you have given the initial state $$\psi_{0}$$ acted on by some operator U and then given the effect of this. Allowing for your inability to explain things properly what the source you've gotten this from was likely talking about was that you can formulate the interactions as an operator on the non-interacting system, this operator being U.
$$0|\Psi>= \frac{1}{\sqrt{2}}[|g>+e^{-i\phi}|g*>]$$
Another example of your inability to understand the source material you're copying. The left hand side as you've written it is zero times some state. Now it doesn't take a whiz to realise that zero times something is
zero, which is not what you have. What the thing you copied likely says is operator O acting on the state, which is entirely different. Clearly you
didn't do this algebra yourself else you'd not have said something so daft as 0 times a state is something non-zero. But then if you'd done the algebra yourself you'd know what an operator and a state and a Hilbert space is and you'd not have used anything to do with superspace (which you haven't mentioned past your initial definitions so its completely irrelevant).
So far, we consider the mixed state $$\alpha_0 \chi \phi* + \alpha \chi \phi_{+}*$$ consisting of an abstraction of a particles spin description and that of a virtual particle description in a state of entanglement.
Putting on a white coat doesn't make you a scientist, just like you mouthing stuff you lifted from other sources doesn't make you a physicist.
So far, we consider the mixed state $$\alpha_0 \chi \phi* + \alpha \chi \phi_{+}*$$ consisting of an abstraction of a particles spin description and that of a virtual particle description in a state of entanglement.
This means $$\chi_{\pm}= \alpha_1[(\begin{pmatrix} 1 \\ 0 \end{pmatrix})]+\alpha_2([\begin{pmatrix} 0 \\ 1 \end{pmatrix}])$$
That's not an entangled system. Your expression for $$\chi_{\pm}$$ is nothing more than a general linear combination of a 2 dimensional system. Using the notation $$\chi_{\pm}$$ implies there's something specific about each case but the right hand side doesn't have any $$\pm$$ in it and its not a specific system, its the general expression for
any state in that Hilbert space. You haven't done anything, your conclusion is a specific example of the assumption you made further up, $$|\Phi\rangle = \sum c_{n}\psi_{n}$$, which itself is a property of the Hilbert space formalism. You've done in a massive pointless circle and done nothing to do with entanglement.
You
are Reiku2.0, either literally or in spirit, in that he does precisely the same as you, lifting (sometimes wholesale) large sections of detailed work other people have done and then trying to tweak it so that it can't be easily found by doing a Google search. Of course in doing that tweaking he (and you) inevitably introduces mistakes because h doesn't understand what he's altering or how he should alter it. You have obviously lifted much of what you've posted in part or in whole from other sources because you have absolutely no reason to mention superspace nor do you make any use of it. You have misused basic terminology which should be learnt before ever even opening a QM book.
How many times are we going to go through this, where you copy stuff incorrectly or irrelevantly, you get your deception exposed and then you cry about how we're all so mean? No one here forced you to post shit, so you can't complain that its me being unkind or being crude, you have
lied and its laid out for everyone to see. Why on Earth would you come to a physics website which you know is read by people who do quantum mechanics for a living and then try to bullshit your way past them claiming to be doing your own QM research? No rational person would try such a thing and
certainly no rational person would try it
repeatedly as you have. What drives you to do it? Desperate for attention? Extremely stupidity? A potent combination of the two?
No doubt you'll whine about how mean I've been, you might even start another whiny thread over in the feedback forum to complain about how low class 'certain members' (ie me) are but I'm also sure you won't be able to retort the criticisms I've made of your post because you know full well you've plagiarised, copy/edited and just bullshitted that post together.