What does it mean to be Covariant?

Hows it out of place? How is it spurious, explain how it's spurious? I know what is meant by it, so..?

Secondly, this is not what this thread is about; before I was accused for lying by yourself and guest, I have shown consistently to know what a frikken Hilbert Space is. It's not my fault you cannot back up your claims.

 

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Really? Go on, I'll give you another chance....

Here you confused states and operators. You also fail to understand what properties trivially follow from a Hilbert space. Nowhere have I seen you show a working understanding of what a Hilbert space is, so the evidence is you consistently have shown you don't. Its not my fault you can't back up your claims. I keep giving you the opportunity, like giving you a chance to alter what Guest has quoted you saying, but you never step up.

 

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On the contrary. You've demonstrated that you lack even the understanding needed to paraphrase meaningfully! But please, continue to prove my point...

What does this mean?

 

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You really don't. Not more than a week or two ago, you didn't even know set builder notation, nor how to delimit simple intervals on the real line. I cannot even remember when I was taught the latter, but I believe the former was taught in maths class as the very first thing in my first year in high school (gymnasium). 3 years before I went to university. Where Hilbert spaces was 2nd year material.

I flat out refuse to believe that you picked up the 3 or 4 years of mathematical training you need to get to Hilbert spaces from your current (rudimentary) proficiency level in less than two weeks. It's impossible.

So you're simply lying. Again. And because we know that you don't know, that means that your post on the definition of Hilbert spaces is by necessity garbled plagiarism. Again. And to top it off, you even have the gall to report others for calling you out!

I predict that you're not long for this forum.

 

No, it doesn't.

It's perfectly obvious from that post that you don't know what a Hilbert space is: You didn't even explain the two operative terms in the definition: complete space, and inner product space. It degenerates into trivialities and nonsense from there.

As has been pointed out numerous times: You're a fraud. You borrow the terminology and pretensions of science, but everything you post is devoid of understanding or insight.

 

Look again:

There's something in this sentence that's very strange if one is familiar with the notation and nomenclature. What is it?

Pot. Kettle. Black.

Sometimes the best defense is retreat. You're being massacred and rather than live to fight another day, you continue throwing pointy sticks at the advancing tanks.

 

Thread closed.

 

Reopened by request.

 

Whose request? Certainly not mine Ben.

 

It was on my request - I want you to answer my question.

 

In Cauchy Sequences for finite space and infinite space we use the notation \((\ell_2, L_2)\) if that is what you are referring to.

 

I'm afraid that doesn't make any sense. Care to try again?

 

Well, you are being fascinatingly vague. Perhaps if you were to point out the incongruity we might be able to proceed further.

 

Perhaps you could explain what \(\langle \psi | \psi \rangle\) actually is?

 

I wonder what I was talking about.

I was speaking about the norm of a vector wasn't I? By memory I can say that you define the definate norm as \(||\psi||=^{def} <\psi|\psi>^{\frac{1}{2}}\) Which is the positive square root of the positive \(||\psi||\) side. I'm not aware of what is wrong with this?

 

For instance, I assume these things are well-known, or atleast I hope because I have committed it to memory.

For example, knowing such relationships to the probability amplitudes, you can state that \(|\psi>=a-ib\) and \(<\psi|=a+ib\) where \(<\psi|\psi>=a^2+b^2\) making a real positive number.

 

Ok, that's a good effort. Some slight misunderstandings though; \(| \psi \rangle\) and \(\langle \psi |\) are, as you imply, vectors (or to be sightly more pedantic since they abstract as opposed to "real" vectors like force - they are elements of a vector space) and you seems to have equated them to the complex numbers \(a \pm i b\). Can you please explain what you mean?

Maybe it would be enlightening if you wrote down \(\langle \psi | \psi \rangle\) in the language of wave mechanics. That way it's pretty clear what you are physically talking about.

 

That was just an example, I never thought about expressing it any other way. As for wave mechanics, what did you have in mind? I thought maybe \(|\psi>=exp(-ipx)\) would suffice, or maybe \(|\psi>=e^{i(kx-\omega t)}\) for a wave mechanical approach?

 

Or since I have defined this as vector spaces, I could say \(<\psi| \in V\) and \(|\psi> \in \dot{V}\) is it's dual.

 

GD has read the definition of \(\langle \psi | = (|\psi\rangle)^{\dag}\), ie the "Take its conjugate" definition, and in an attempt to look like he knows what he's doing he's go to the only bit of complex analysis he knows and done the \(zz* = |z|^{2}\) formula, probably motivated by the mention of amplitude in the explanation of wave function norms. Hence why he seems to be mixing things up, equating a complex amplitude with a state.

What an odd thing to say, since if you'd read books or lecture notes on this stuff you'd have been people do things differently.