What does it mean to be Covariant?

[prometheus]

So expressing that in coordinate space can be as trivial as $$<\psi|\psi>= \int <\psi|x> <x|\psi>d^3x= \int \psi^{\dagger}(x) \psi(x) d^3x$$

Excellent! So what is $$\langle \psi | \psi \rangle $$ physically?

 

[Guest254]

Oh, I shouldn't have the infinity sign should I, in a finite $$\ell_2$$ space. Is that the spurious thing which was meant?

$$\ell^2$$ is infinite dimensional. Keep digging...

 

[Guest254]

Of course I've seen it before, it's used for a discrete basis state in this case for position so the completeness is given as $$\mathbb{1}=\int |x><x|d^3x$$.

Congratulations once again - the spectrum of $$\hat{x}$$ is continuous (i.e. not discrete).

 

[funkstar]

I came here as a novice, made that clear, but it still fell on deaf ears, trying to state I made it out to be more.

You certainly did not make that clear until you had been caught out multiple times. And then you continued your intellectual impostures, instantly negating any sympathy or leeway might have been offered you on account of being a layman.

As I said, I am sick to teeth about these lynching mobs.

You're being called out on not knowing the basics, while claiming that you know them. This happens all the time in any educational system, and life in general, and shouldn't be unfamiliar to you. When you got red marks on your maths exercises in school did you then feel that the grader was "out to get you"?

Even funkstar is trying to make it out you need to know a great deal more about Hilbert Space to say you know what it is - I disagree - I'm not going to start teaching on them, because yes, then I would need to know how the metric fit in to it, but to be honest, I don't think it would be very hard to find out how.

You simply don't have the expertise to disagree. In my best assessment of your current competence level, based on the posts of yours I have read, you have years of material to cover before you could credibly claim to know Hilbert spaces.

Imagine that this was a message board on medicine, and that you had claimed to "know how to perform an appendectomy". However, when asked about details, your knowledge was essentially limited to "cut open the abdomen, remove the thingy". Would you really be surprised if other posters were to attack your claim of knowing how to perform an appendectomy?

Now, does your ignorance mean you should stop asking questions, here or elsewhere? No, of course not - asking questions is a central part of all learning. But understand that until you know the mathematical framework, your questions on anything beyond your comfort zone (i.e., those thing you know), and any, to you, useful answers will have to be simplifications, largely hoping to convey intuition rather than detail, usually with severe omissions.

 

[Green Destiny]

Why thje trivia guys? I don't have the patience for this.

Guest, stop fucking dancing round the bush, and just tell me where your problem consists.

 

[Green Destiny]

Congratulations once again - the spectrum of $$\hat{x}$$ is continuous (i.e. not discrete).

Oops/.

Yes, you're right. I got mixed up with $$\sum_n|x><x|= \mathbb{1}$$ which is of course, discrete, the above is continuous as you said.

 

[CptBork]

Ok Green Destiny, let's give you a nice easy test so you can at least show us you understand basic vector spaces, as you claim.

Consider the following four vectors:
$$\begin{pmatrix}1\\0\\0\\0\end{pmatrix} \begin{pmatrix}0\\1\\0\\1\end{pmatrix} \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}\begin{pmatrix}3\\0\\5\\1\end{pmatrix}$$

Find a set of basis vectors for the vector space spanned by the vectors above, with the set of scalar multipliers being defined as the set of real numbers.

 

[Green Destiny]

Ok Green Destiny, let's give you a nice easy test so you can at least show us you understand basic vector spaces, as you claim.

Consider the following four vectors:
$$\begin{pmatrix}1\\0\\0\\0\end{pmatrix} \begin{pmatrix}0\\1\\0\\1\end{pmatrix} \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}\begin{pmatrix}3\\0\\5\\1\end{pmatrix}$$

Find a set of basis vectors for the vector space spanned by the vectors above, with the set of scalar multipliers being defined as the set of real numbers.

I'm not home, so I will need to do this from whatever memory I have left.

Is it ok to ask questions?

 

[Green Destiny]

My first question, is just something I want confirmed. To find your basis vectors in a vector space, you find the maximum linearly independant vectors yes?

 

[Green Destiny]

I suppose I would start by defining the matrices you gave in the correct fashion, by stating:

$$x,y,z,t= A \begin{pmatrix}1\\0\\0\\0\end{pmatrix} B \begin{pmatrix}0\\1\\0\\1\end{pmatrix}C \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}D \begin{pmatrix}3\\0\\5\\1\end{pmatrix}$$

So

$$\begin{pmatrix} x \\ y \\ z \\ t \end{pmatrix}= (AV_1,BV_2,CV_3,DV_3)$$

Would that be right? I am a bit weary.

 

[Green Destiny]

I suppose I would start by defining the matrices you gave in the correct fashion, by stating:

$$x,y,z,t= A \begin{pmatrix}1\\0\\0\\0\end{pmatrix} B \begin{pmatrix}0\\1\\0\\1\end{pmatrix}C \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}D \begin{pmatrix}3\\0\\5\\1\end{pmatrix}$$

So

$$\begin{pmatrix} x \\ y \\ z \\ t \end{pmatrix}= (AV_1,BV_2,CV_3,DV_3)$$

Would that be right? I am a bit weary.

I'll just keep going, and maybe you could point out if I am wrong.

From here, my memory recollects that:

$$A \begin{pmatrix}1\\0\\0\\0\end{pmatrix} B \begin{pmatrix}0\\1\\0\\1\end{pmatrix}C \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}D \begin{pmatrix}3\\0\\5\\1\end{pmatrix}= \begin{pmatrix}0\\0\\0\\0\end{pmatrix}$$

For some reason, I recall setting the vector on the right to being zero in all column entries yes? Am I along the right track?

 

[Green Destiny]

mistake, be right back

 

[CptBork]

You just need to find a set of linearly independent basis vectors for the vector space I defined above, that's all I'm asking for.

 

[Green Destiny]

You just need to find a set of linearly independent basis vectors for the vector space I defined above, that's all I'm asking for.

That's what I was attempting to do. Since I am not home, I am having to look back on previous posts I have written:

http://www.sciforums.com/showthread.php?t=69371

 

[CptBork]

Hint: It's so easy, I've practically already given you the answer right here.

 

[Green Destiny]

This is the only way I know of doing it. You need to write out simultaneous equations for:

$$A \begin{pmatrix}1\\0\\0\\0\end{pmatrix} B \begin{pmatrix}0\\1\\0\\1\end{pmatrix}C \begin{pmatrix}0\\1\\0\\-1\end{pmatrix}D \begin{pmatrix}3\\0\\5\\1\end{pmatrix}= \begin{pmatrix}0\\0\\0\\0\end{pmatrix}$$

is that right?

 

[Green Destiny]

This matrix algebra latex thingy is screwing with my head.

 

[CptBork]

Hint 2: If you knew anything whatsoever about vector spaces, this is one of the first things you would have learned. It's not complicated at all, I gave you a very simple vector space and all you have to do is give me one basis for it out of the infinitely many choices you could make; there's even an obvious one that's been practically staring you in the face this whole time.

 

[Green Destiny]

I am not taking part in this anymore. I have been very civil, explaining what I know. If you are going to resort to an attitude like that (I expected more from you) then I will not take part.

I have explained I have never taken a class in these subjects, so to identify the ''thing'' you are wanting was down a certain path I learned to find them. If you are going to be awkward, forget it.

I never asked for this trivia, I am surprised I've returned to this thread as many times as I have.

 

[CptBork]

If you don't know what a basis set is, you can't tell me you understand more than the bare minimum about vector spaces. Many of the most basic results in linear algebra rely on the concept of a vector space basis. You can choose to cop out now and keep this thread on record with you giving up, so we can point to it every time you claim to have some understanding of a vastly more sophisticated topic, or you can finish what you started and answer pretty much one of the simplest vector space questions a guy could ever ask you.

I don't get why this is such a big deal. No one would hold it against you to say you haven't learned a particular area of math, or that you attempted to study it and later realized your perceived understanding was completely mistaken. What do you expect to gain by trying to discuss math subjects you're not ready to discuss?