studiot

Are you going to listen ? Or are you just going to continue trotting out your misconceptions ? A space is a container for whatever I want to put in it. Common spaces include vector spaces, coordinate spaces, phase spaces, topological spaces....... the list goes on and on. Usually a space contains at least one set of objects, and a set of rules. So a vector space contains a set of vectors, a set of coefficients, a set of rules, perhaps a set or sets of the results of those rules. Listen and learn and you will find your discussion with others so much more rewarding. You are often partly right in what you say, but you are missing out on so much. For example This is known as a sufficient but not necessary condition. Yes that is one way for us to observe the particle behaving lik A. However there exists at least one other way, quite independent of the particle or even the existence oft he particle. If we set up the observation to observe only A that then is what we may or may not observe. But when we do it clearly implies the existence of the particle and its action as A. The real fun in quantum or any other theory starts when we do not observe A.

https://www.bbc.co.uk/news/uk-scotland-63402811 Just look how low this dam is.

1) I thought the definition of 'space' would need clarifying at some point. You need to understand something they never teach you at school. The 'space' defined by the position is different from the 'space' that encodes the quantity itself. When explaining this further I like to illustrate it with a piece of graph paper and the addition of vectors by a vector triangle or parallelogram. 2) This he said-she said argument could go on indefinitely. So hereare two treatments of how to implement this mathematically. Both are fully compatible with each other and explain what others have been saying, including sensei's compound scalar fields. The first come from Marsden and Tromba Vector Calculus The second from Schwartz, Green and Rutledge Vector Analysis

I too have been trying to introduce a measure of levity to this discussion. Around 150 years of exacting spectroscopic measurements and the development of the corresponding quantum theory would say otherwise. Spectroscopic theory is about the most complete and accurate that we posses.

Really ? I find the rudeness of the dismissal the most suprising part in your reply to someone who has offered sources of genuine help for your consideration. And this is not even your thread. You clearly do not understand the most fundamental point about entanglement at all. The entangled properties are set for both particles at the moment of entanglement. In order to create entanglement you require very close proximity - not spooky action at a distance. But once set it does not matter how far they diverge, anyone who measure one automatically knows the other. I'm a little hydrogen atom, You're a little hydrogen atom I've got one electron You've got one electron Let's get spin entangled and form a hydrogen molecule. At this point we know that one electron is spin up and the other is spin down But we don't know which is which until we observe one of them. Thanks to joigus +1 for trying to explain my sine example and providing another example.

You didn't reply to my last post, but I will try this anyway. Consider the equation sinx = 0.866... No one is suprised when we say that the solution to this equation is x = 60o or x = 120o or x = 420o or a further infinity of solutions. So are all these solutions in superposition or entangled, and just waiting to be selected or observed ?

I'd say this is the key to your misunderstanding I am taking this as a reference to the mathematical definition of a Field as a set with two binary operations obeying the 10 Field Axioms, F1 - F10. Two of the axioms refer to algebraic closure with I take to be the formal statement of the words in your quote I have emboldened. I'm sure you can appreciat that this is a far cry from the Physics definition (taken from a book of vector calculus) Note the mathematical description of the scalar field may be more or less complicated than the mathematical description of some vector field. for example (in 2D to make it easy) Scalar field S(x,y) where r is a constant such that x2 + y2 = r2 Vector field u(x,y) = (y,x) (Note I would be interested to see sensei break this down to two or more scalar fields)

Once again I can partly agree with your statement. Physics does indeed 'use' logic. But logic is not the basis or foundation of Physics, nor is it the final arbiter of any proposition in Physics. If you do not understand and appreciate this it may be why you are having conceptual difficulties in Physics. By the way you did not actually answer as to whether you appreciate and understand the difference between a Field in logic and a Field in Physics. This is fundamental since you startd by insisting that Fields have substance ("they must be made of something"). Nothing in logic has substance. Plase note these points are meant to help and make you think it out for yourself.

Since you don't want to talk to me or answer my legitimate question, have a nice argument with swansont and joigus.

This seems to be a rather mixed up point of view although I would agree that "the idea that a field is numbers assigned to space points. " is indeed the beginnings of the way Physics regards a Field, but there is much more to it that that and the definition of a Field can be made much more complete and useful. I also find it interesting that your previous posts have been about formal Logic as in Mathematics. Here a field has a totally different unrelated definition, mcuh less 'natural' than the Physics idea. How do you view this ?

Like it. +1

Newton's Bucket ? https://en.wikipedia.org/wiki/Bucket_argument By the way that's a crap picture you posted, quite unrepresentative of even the simplest classical mechanics taught in junior high school.

My local public library is very good at offering new books. You might like to take a look at this one that has just been published. https://www.amazon.co.uk/Impossible-Possible-Improbable-Science-Stranger/dp/1785788825 The book is divided into the sections, as per the title. The first section deals with QM and in particular interpretations of QM from Copenhagen right up to the present day. Localism, realism, Bells, Hidden variables and many famous names are all discussed. Of particular interest are six different interpretations of QM, superposition, entanglement, etc. Here is the expanded contents list for this first section.

Rather than wade through a lot of organic chemistry that is of little interest to you why not try something like the latest version of this ? Other tnagential approaches might be the Oxford books on Physical Chemistry for Biochemists and Pharmaceutical Chemistry. Come back if you want more details.

Although a true statement, I have absolutely no idea what this means here or is relevent to.

I realise that English is not your first language but your English is pretty good all the same. So I am suprised how completely you have misread my entire post. I am sorry it was not more clear.

Whilst I agree with your sentiment, I can't fully agree with your reasoning. +1 The sentiment expressed in both your posts suggests that we should be (collectively) making much better use of the resources our planet provides. +1 I have been saying this since I first took an interest in the 1960s. But surely the worst offenders against both global pollution and weaponisation are actually also the least democratic nations ? Democracy is undoubtedly a less efficient organisational method in some respects, but it remians the best, perhaps the only, defence we have against totalitarianism. As such, those who enjoy it are prepared to accept the increased 'cost' of deploying it in some form. Finally I think you once mentioned that you are in Poland. Do you not find it ironic that Poland now finds itself better placed to reject Russian gas than Germany, becuase of its insistence on maintaining its brown coal power source ? I also agee with MigL. +1 We need to do two things; one short term one long term. Both of these are consistent with the long term aims of 'making the best uses of our resources' sentiments. But we also need to take more responsibility for our individual and collective actions, and not try to shove them off onto someone else. Sadly I don't see this revolution happening any time soon. Sadly short-termism will hold sway for a long time to come.

Fixed.

I asked you a simple question to which you have not made any reply. All you have done is repeat your early claim more strongly without any support whatsoever. So I will make it eay for you. The waste material from coal burning is called PFA (pulverised fuel ash) or fly ash. Far from being worthless it is a useful industrial material in its own right as it is pozzolanic. To make it easy here is a breakdown of the minerals in PFA https://onlinelibrary.wiley.com/doi/abs/10.1002/jctb.2720240405 The reaction of calcium oxide and carbon dioxide. This will lock up some carbon dioxide, but will not make any progress towards turning the waste into arable soil.

Thank you for the information. The Wiki article you linked to seems to suggest fracturing would be needed.

@Ayub Umar Whilst I think you suggestion will do far more harm than good, it is really good that you are thinking about soil health. The continued reduction of soil health over large swathes of the planet is a hidden danger for us all. https://www.bbc.co.uk/news/business-63283986

The point is we want numbers not hand waving. If you go to your bank manager and say There are lots of cheap bananas and coconuts in the Windward Islands. I want to make money by buying them there, taking them to New York and selling them for much higher prices. He will say What is you business plan ? I want numbers. How much do they cost? how much can you sell (some) of them for? How much wastage and unsold stuff will you have? What are your storage and transports costs? What are you import regulatory costs? What are you staff costs? It's the same with Science and Technology. It's up to you to show the true maths, not offer wishy washy statements like

And I did ask for an example of this 'anything' , whilst at the same time pointing to a well respected treatment of QM, involving plenty of imaginary and/or complex numbers. But this is taking the discussion away from the key statement Yes I am aware of extension algebras and arithmetic and I know tha there are some pretty exotic creations floating around there, but I am not very adept at them. I was always given to understand that for every gain in one direction you tend to loose something in another when you add to algebraic complexity.

Entirely ? I don't think so. But then I don't think you mean quite that literally. Nor are real numbers entirely commutative either. For example multiplication and square rooting are not commutative 2.√3 is not equal to √2.3 They are both commutative for either of the two operations of addition and multiplication as befits their status as Fields, which I believe I mentioned. But QM commutation is about the commutation of two different operators. So can you detail your real operators in QM that do not commute ? The dreaded 'i' seems to appear lots in this treatment. https://quantummechanics.ucsd.edu/ph130a/130_notes/node109.html Maybe the factorisation was stretching things a bit far as composition is involved, but I can't see what you mean by this How is (1+√5)/2 an element of an integer ring, when division is not closed with the integers ?

Actually there is more to what Markus said than meets the eye. The twist in the tail of using complex numbers in QM lies in non cummutativity. The 'full' set of complex numbers entail numbers of the form a + bi, where a and b are real numbers. Both the real numbers and the complex numbers formed this way form an algebraic Field and enjoy all the field properties, including the 10 Field axioms. One of the most important consequences is the unique factorisation theorem which guarantees unique solutions to all alegbraic equations in one of these Fields. Now Gauss discovered and did a lot of work on something simpler, we now call 'Gaussian Integers'. These connect ordinary integers with Gaussian integers in the same way as the reals and the complex numbers. So if p and q are integers then p + qi are gaussian integers. Now neither the ordinary integers nor the gaussian integers form an algebraic Field. They only satisfy 6 of the field axioms. But they do form a more general algebraic structure called a Ring. Some rings such as the integers still satisfy unique factorisation so for instance the only factorisation of -28 is -(1x2x2x7). But in the ring p+q√5i, the number 6+0i factorises as follows:- (2+0i)x(3+0i) and also (1+√5i)(1+√5i) Such rings of complex numbers fail to satisfy unique factorisation - essentially a quantum behaviour.