Dirac Sea

It seems much of the equation was understood when Dirac starting postulating on it - atleast, many of the mathematical rigors. It wasn't until afterwards Pauli came to Dirac with a solution to prevent electrons from simply falling back into the sea, which is precisely what the matrices do.

 

Log in or Sign up to hide all adverts.

You can't do anything quantitative though, which is precisely my point. I asked you to clarify what you meant by 'investigating' and you ignored my question. If you haven't any understanding of the quantitative stuff then all your 'investigation' is just reading wordy explanations other people have provided you and then just making stuff up. Without the quantitative understanding you have no way of developing any results which aren't just baseless conjecture.

So what precisely was your 'investigation'? What did it involve?

See this just demonstrates how you have no intention of not spouting BS. The Pauli exclusion principle was years before QED and the Pauli matrices were used in non-relativistic quantum mechanics before too. Pauli didn't go to Dirac after and say "I've got a way of stopping them falling into that state", Dirac built it into it from the start.

Dirac was after a 1st order (in space-time coordinates) differential equation to describe the electron which would then imply the second order wave equation. Essentially he wanted an operator which could 'square' to the D'Alembert operator. But it had to have spinorial properties, as spinorial fields manifest the Pauli exclusion principle by anticommuting. However, the operator couldn't have vector properties so he couldn't just use the simple case of \(\partial_{\mu}\) as its vector valued. Instead he combined it with a set of matrices whose indices are spinor indices and whose 'family' index is a vector index aka the Dirac matrices, ie \(\partial_{\mu} \to \gamma^{\mu}\partial_{\mu} = D\) where \(\gamma^{\mu}\) has indices \(\gamma^{\mu}_{\alpha\beta}\). If \(D^{2} = -\partial_{t}^{2} + \Delta\) then you need \(\gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu} = \partial^{\nu}\partial_{\nu}\). It follows that \(\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu} \equiv \{\gamma^{\mu},\gamma^{\nu}\} = 2 \mathbb{I}\). This is the defining property of a Clifford algebra.

The Pauli matrices then come into via the use of chirality. In 4 dimensions you can define chirality on a spinor via \(\gamma^{5}\) and thus can split a Dirac spinor into opposite chirality Weyl spinors. The Dirac equation then becomes an equation involving 2 2d spinors and the Dirac matrices for 2d spinors are the Pauli matrices. That is how they arise in the Dirac equation, it wasn't Pauli saying after the work had been done "I've got something you could add in", they follow from the essential property of the Dirac operator. The Pauli matrices weren't an afterthought, Dirac knew what he needed to put in to build up the right mathematical structure and the Pauli matrices are implicit in the methods he used.

 

Log in or Sign up to hide all adverts.

I attempted at finding different but reliable resources which helped to explain the Dirac Equation. Until I began to find information repeating itself was when I realized I was pretty much finding all the prelimineries required.

Don't be too harsh as I don't know why I formed that opinion. I just did my history there, and you are right. Apparently Dirac classed his equation of independantly discovering 2x2 matrices and pauli independantly discovered his.

 

Log in or Sign up to hide all adverts.

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.

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 \(H\).

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 \(S_2\).

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 \(\phi*\).

If system \(S_1\) was created from the same source as \(S_1\), then they belong as a corrolated system. The eignstates of the mixed-system represented by the supserposition of \(S_1+S_2\) would then be given as:

\(\alpha_0 \chi \phi* \mathbb{V} \alpha \chi \phi\)

where \(V\) denotes the ''or'' and \(\alpha, \alpha_0\) are complex coefficients. The joint system therefore has an initial state given as:

\(\psi_0= \chi_0(\alpha_0,\phi*+\alpha,\phi)\)

We therefore have a joint state simply as \(\chi_0,\phi*\) and \(\chi,\phi\). 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:

\(|\Psi>=\sum_{n=1}^{\infty} \psi_n c_n\)

If the range of the \(n^{th}\) power is effected by a noisy thermal background (or some kind of strong coupling on a system) on the system \(\mathbb{S}=(S_1,S_2)\) of subsystems, their interaction terms \(U\) can be given as:

\(U\psi_0= \alpha_0 \chi_0 \phi*+\alpha \chi \phi\)

In our case, we will use a ground state condition for the state vector on the Hilbert Space,

\(0|\Psi>= \frac{1}{\sqrt{2}}[|g>+e^{-i\phi}|g*>]\)

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 \(\chi_{\pm}= \alpha_1[(\begin{pmatrix} 1 \\ 0 \end{pmatrix})]+\alpha_2([\begin{pmatrix} 0 \\ 1 \end{pmatrix}])\)

 

How long did it take you to come up with this?

 

In other words you're going to just make stuff up or lift it from elsewhere and hope people swallow it.

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.

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 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.

Systems don't have eigenstates, operators do. You are demonstrating you don't understand basic terminology. Again.

The what?

Spurious use of commas.

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!

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!

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.

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).

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.

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.

 

I never copied anything. I am trying to remember what you call ''basic stuff.''

 

One night.

 

You don't even know how to delimit a basic interval, yet we're supposed to believe you have any competency in this stuff? I'm sorry, but I cannot suspend that much disbelief - The Core gets more things right than you do.

This is such an obvious lie Shaggy would be embarrassed.

You cannot remember what you have not committed to memory. And you never learned any of this stuff. AN has it exactly right - you copied something you don't understand, and smothered it in technical terms you also don't understand, and are now trying to pass it off as your internalized knowledge.

You're a fraud, plain and simple.

 

Ok.

You don't have to like me. I understand that.

 

Anyway, Alphanumeric has assumed some things not true to the work, because it wasn't explained in full detail. The wave function will be non-linear in evolution, despite what it seemingly shows because of a strong coupling on the system. The wave function of probability will follow rules similar to

\(\Psi(t)=sin^2(\frac{\pi}{2}t)\)

and so \(N: \Psi \frac{1}{N} \ne \Psi(1)\)..

Which will require that there is a change in the probability field which is not a linear state in time.

 

Rubbish.

The prime example of why you're talking rubbish is your mentioning of superspace. You'll be hard pressed to find an undergraduate course in physics which covers superspace, as its a concept which is not introduced until a student has done quantum field theory and wishes to go further into it, ie supersymmetry.

If you'd done superspace you'd know it has nothing to do with the 'basic stuff'. You'd know that you make no use of it after you mention it. You'd know that it has nothing to do with the point you were trying to make.

What about all the other things I point out, where you misuse terminology, get definitions wrong and quote irrelevant things?

If you were really someone honestly trying to do physics after a few years and were just a bit rusty you'd have not worked all the way through to a conclusion and then only said "Oh I might be mistaken" after someone points out all your errors. You would have instead discussed things with people and gone one step at a time. I'm absolutely certain that if I (or someone else) hadn't gone through all your errors you'd be quite happy to say "This is all correct, see I can do quantum physics" (which is precisely the attitude you've taken in previous threads).

You clearly aren't trying to remember basic things because you never learnt the basic things in the first place and since you're then forced to copy/edit other people's work you don't know where the line between 'basic terminology' and 'advanced research' is.

Are you admitting to being a fraud? If not then explain why you mentioned superspace. Explain why your 'conclusion' is nothing but a stating of a definition. Explain why you misuse terminology about Hilbert spaces. Explain why you can't multiple something by zero properly. Explain why your posts smack of you copying other people's work and trying to edit them enough to look different.

Precisely what have assumed? Precisely what have I criticised which I shouldn't have?

And let's consider your new attempt to dig yourself out of a hole of your own making....

You haven't given any interaction potential. I previously commented you had mucked up that bit of your post.

Looks like someone has been looking up Xeno's paradox in quantum mechanics. You once again demonstrate you don't understand what you're saying because you don't realise that that particular solution is to a specific equation. A wavefunction can be anything which satisfies the Schrodinger equation. If you are talking about a specific potential then you're talking about a specific form of the Schrodinger equation and thus the wavefunction from one system will not be valid for another. Since you haven't given the potential and you actually mention 'strongly coupled' (but don't define what you mean) you can't then just pull out a wavefunction which happens to come from a very simple system (its simple enough to be given as homework to QM students during their first course) and talk about it.

Yes, the quantum mechanical concept of "a watched pot never boils" needs for the wavefunction to not be linear in time. What precisely does that have to do with anything in your previous post, which was (supposedly) about entanglement, a completely different phenomenon?

Precisely as I predicted, you complain but you don't actually retort any criticism made of you. Instead you spout more shit and hope somehow it'll mean your previous posts weren't also shit.

Rather than spouting more liars or more stuff you copied from elsewhere why don't you try having an honest discussion? Or is having to admit you're not a whiz at physics too much to ask of you?

 

I needed an interaction, other than one involving simply a collapse model.

And I thought superspace was basically the space configurations of possibilities. I was obviously wrong.

 

Collapse model? What collapse model? The wavefunction collapse is not a model of a collapsing system, the dynamics of the wavefunction are quite different from the notion of it collapsing. Collapsing is done using observations, the dynamics are found using the equation of motion, the Schrodinger equation.

Why didn't you look it up? If you're trying to remember why aren't you checking your work first? Why aren't any of your posts phrased as "I had a vague idea which I think involves the following", why are they all phrased as "Here's the details you wanted which demonstrate I'm right", which then turn out to be bullshit? There wasn't a hint of "I'm a bit rusty" in your reply, it was clearly meant as "I've managed pairs of pairs, here's my derivation of that, as you asked", as if you had a firm belief you were presenting viable science.

This is not the first thread, you made a similar slew of basic errors, both mathematical and in terminology, in your electromagnetism thread. In that
thread you used similar excuses, about how you're a bit rusty. Clearly you learnt nothing from that as you behaved exactly the same here, spewing out crap as if its viable and then making excuses when you get called on it. A reasonable person would have realised that if they've forgotten so much then perhaps they should read up on it before trying to pass it off as science.

And I can't help but notice you're ignoring my questions. Can't you explain why your final equation have nothing to do with entanglement? Can't you explain why you don't check anything you say before you say it? Can't you respond to my question about you being a fraud? Rather than spend an evening (so you claim) writing another BS thread like this one how about you spend that time answering questions in this thread which people ask you? If you're unwilling to defend yourself when you post BS and someone calls you on it don't be surprised that the next time you post BS people treat you even less pleasantly, it is seems like you're avoiding acknowledging your lies and mistakes and you just want to keep pumping out more lies and mistakes in the hopes someone might think you're not a thick lying hack.

The problem is that your style is known and no one is going to give you the benefit of the doubt now. Instead of thinking "I don't know this but he seems to be using lots of big words so he's probably right." the default image of you now is "I don't know this but he seems to be using lots of big words so he's probably making shit up.". Time for a new username Lee.

 

To be fair, half your post consists of accusations rather than real questions, hence I intentionally miss them out, because to be frank, I have a wicked hangover, and I just don't have any motivation for you today. Or most days.

 

Enough time to write crap but not enough to defend it? Hallmarks of a hack.

 

Tell that to the bottle of vodka I punished last night.

The bottle might care more. :shrug:

 

So that's your excuse for today, what about all the other days? what's your excuse for never being able to defend yourself against criticism with anything other than more bs or saying you 'forgot' the right expression.

I asked you what I incorrectly assumed, so you claim, about the opening post. How about you give specifics rather than whiny excuses?

 

Wow.

You're actually begging me nearly to answer you. I never realized you looked so forward to our little rendezvous.

 

yeh, feeling better now. How are you?

In fact, don't bother answering, I really don't care.

Anyway, I will cut to chase. I read your post again to rephresh my memory, and alas, I am still reminded to why I keep ignoring you. Until you work on those communicational skills, more truthful questions and less decieving tactics, like ''sock puppet allegations,'' I will not be part of it.