Toward General Theories and Extensions I

In these next few posts we want to try and salvage some sort of a general statement about acceleration.

What we will discover is that we can discuss the nature of spacetime, if we start from these basic ideas, without cluttering the discussion with unnecessary mathematical machinery.

We will of-course, GOD willing, discuss that same machinery later on. But notice that we haven't explicitly touched upon aspects of symplectic geometry as yet. This by no means makes our discussions meaningless or exceedingly simplistic. In the meantime, we sacrifice mathematical machinery for a clear understanding of the spacetime events.

On Acceleration III

On Acceleration II

  

   

On Acceleration

In these next postings, what we try to do, we extend the usual standard classical models. In particular, we ape the notion of a coordinate acceleration. That which is equivalent to the difference between the proper acceleration and the acceleration in a frame in which a particle/entity is undergoing constant acceleration.

 The link to this rather tenous example has been provided above.

Imagine that the force/acceleration exerted by a particle M^ (M hat) onto another particle U (mu) changes as M^ approaches U.

In the next frame, imagine that M^ is as a result of the breakdown of a much larger particle. That M^ is the difference between the larger mass M1 and the smaller mass M*.

What we try to develop is a classical theory of how the change in acceleration experienced by U may develop. Essentially, by replacing the mass M^ by its equality. The reason why we may want to do this is so that we explore how generalisations can be made.

A Response To John

Hi John,

Thanks so much for your comments, I'm sorry for the late response. For some reason I didn't receive any notification of your message on Facebook...but anyway...here goes.

"You write two Ls for independent systems then define some relationship between the systems (in your case 2) then try and solve for the Hamiltonian of one system in terms of the other."

I think that if we were to write a single Lagrangian with the potential GMm/r, we are effectively assuming an interaction between two particles. What I mean is that this potential can be read as GMm/r = mv^2. A physical intepretation of this is that particle M gives, or more precisely assigns, a velocity v to particle m. The potential is itself the result of the interaction between the two particles. Best illustrated if M = 0. The potential is an interaction term. When we talk of Lagrangians I think that we cease to speak of independent particles or systems. We find it probably more appropriate to speak of single pairs of systems.

A single particle/wave/uniform field on its own at infinity does not have a velocity to speak of, nor even a coordinate system of its own. (When I say no coordinate system, I mean none that it can identify. The idea of a particle at infinity is defined by the observer who imagines that there is a particle at infinity, not by the particle itself.)

" This approach assume apriori knowledge of the system dynamics and will not result in meaningful physics in general."

I agree, in the sense that there is a problem with using Lagrangians, higlighted here, because the mass M of the observer must be measured in order to determine the size of the potential of particle m. But this is the nature of gravity, at least Newtonian gravity (though I think the potential g_ab(x) has the same problem/solution) and perhaps all other fields...if you measure the value of the classical field at one point then this provides information for the values of the field at other points.

So there is in a sense apriori knowledge that can be obtained from measuring classical ( and quantum?) fields. But the physics is meaningful. In other words, even at a quantum level, aspects of this method of extrapolition persist, if gravity is accurately described using fields.

The mass M is usually defined as a system of particles that can perform a measurement, the measuring device, on a particle m.

"A more consistent approach includes defining the system to include both particles. The Lagrangian will now include (if you desire) a term that addresses the interaction between the two systems. Now solve the system accordingly."

If we consider the above, that the potential in the Lagrangian contains information about an external observer, then we immediately realise that the potential is an appropriate term. For example, a change in the value of M reflects as a change in the potential of m, according to M. Before we start imagining problems relating to instantenous/action at a distance ideas, we must realise that the Lagrangian is written by the observer - not by the particle itself.

" In quantum field theory the systems to which I refer are called fields and they are not discrete entities as in your example but are present throughout spacetime. Additionally, take care when you say that "Spacetime is as a consequence of observation". A more accurate statement would be to say that human observation make our understanding of spacetime possible."

Up to this point we haven't started addressing the issue of quantization. We haven't even properly addressed the four-component forms and suitable tensors which are to be quantized. We have found an interpretation and used that to discuss the Lagrangian.

"Spacetime is as a consequence of observation". I couldn't find where I said this? If I said it, I'm obviously mistaken or have been presumptious.

"A more accurate statement would be to say that human observation make our understanding of spacetime possible." I agree.

Sizzling London and...Banach Spaces.

It's been sizzling in London...so here's some essential equipment. That's water...real H2O.

I'd appreciate it if someone could tell me what Banach algebra/spaces are...I mean, I know that Hilbert spaces are a type of Banach space, but I really need some introductory text on this and how they relate to Lie algebra (groups).

Recumbent and friend..

I think this thing - called a recumbent? - is really useful. For short fun cycles, not long trips though.

And for hot summer days is there a more leisurely Primani-type shirt?

See Part II: First Adventure and Physical Reality (ii)

See Part II: First Adventure and Physical Reality (i)

With capital M for mass.

Part II: First Adventure (v)

If we ask whether we can use this method for generalised coordinates the answer is yes. As long as we can perform general linear transformations on those coordinates.




So this is not dependent upon the type of coordinate system used but only upon the ability to perform linear transformations.

We, therefore, have a unique presentation of the Hamiltonian. We can try to further generalise this idea by guessing that the radii are components of a metric as shown below.

This gives something that looks, perhaps, background free. Does this suggest that the Hamiltonian must then become a matrix of some sort? Perhaps the presence of the exponent implies this as well as the fact that the radii are components of the metric which is a higher structure, a tensor.



Overall...to where does this lead us. What physical scenarios might make use of all this that we have been suggesting?

See Part II: First Adventure and Physical Reality.

Part II: First Adventure (iv)

We can imagine that our two Hamiltonian are in some sense 'entangled', in this case, purely as a consequence of the type of coordinate system available to us.

We want do this because ultimately itwill give us an insight, or different viewpoint, into particle interaction. We develop the idea as shown...

Part II: First Adventure (iii)

In this case we have substituted by replacing the rest mass with partial derivatives of the Lagrangian. Meaning that we remove any direct reference to rest mass (m instead of 'mu') in our formulation. We can do this in our case without implying that there are other particles in the neighbourhood.

Let's then proceed to writing the Hamiltonians for the mass taking advantage of the implications of our NPM by forming two Hamiltonian instead of just the one.

Part II: First Adventure (ii)

Using the usual method of building the Hamiltonian by taking the derivatives of the Lagrangians - we now have 2 Lagrangians - we define as shown.

Where taking derivatives of the exponent produces a null result because the exponent does not contain any relevant variables.

Part II: First Adventure (i)

In Part I of the first adventure we discussed how it was possible to describe the Lagrangian in our coordinate sytsem (which we labelled NPM/NCM...whatever)



This coordinate system allowed us to interpret the Lagrangian used by Chung et al. as merely emerging from some coordinate system which we labelled NPM/NCM.



It turns out that this coordinate system provides for us a platform upon which we can build interesting Lagrangians. We can now move on to Part II of this our explorative, creative and (add own descriptive) ... tryst.

Are science and religion NOT compatible? A quick comment...

The topic I am commenting on is here http://blogs.discovermagazine.com/cosmicvariance/2009/06/23/science-and-religion-are-not-compatible/

A physicist commenting on religion, if he is not religious, is like a physicist commenting on linguistics. Something of which he has only a peripheral grasp of.

I have heard L. Susskind, whom I respect very much as a scientist, off-handedly say something like

If you put electrons in a box.....

Implying you can test for the existence of GOD by examining the anomalous behaviour of electrons in a box. Which is nonsense. Why would GOD prove HIS own existence? I think that HE is not needy. It's like someone in Africa who has never heard of me, and has no means with which to do so for himself, challenging the fact of my existence. Ludicrous.

* I don't know if Prof. Susskind recanted his comments. *

The reason for faith, as I understand it, is because mankind does not have the tools to prove the existence of GOD. And so, I suppose, until mankind can explore the very structure of spacetime looking for the existence of other dimensions in which GOD may also be found, for now, you either take faith or you leave it. Your choice, apparently with consequences.

There isn't a man alive on Earth today who can prove that GOD exists, therefore, necessarily disprove that HE exists. ( Yes, I mean precisely what I've just said. If you can prove that an infinite being exists then that being is not infinite i.e has a definable existence.) But there are many who become puffed up by the little that they know and like the Emperor with no clothes, display their ignorance.

But in the meantime, whilst we wait for that man, what do you do? Because apparently the decision is not without consequences.

from Lagrangian Pairs III to stuff

Lagrangian Pairs III

What we have done so
far is to follow the standard procedure used to define derivatives of the Lagrangian. In this example, we show how to construct the Hamiltonian using the derivatives as terms.

In our case, we are obliged to define 2 separate but, in a sense, interacting Lagrangians.

The argument for this pairing up is trivial. In the absence of any observer there is no velocity and no spatial or temporal extent.

A particle 'alone' in some void cannot experience spacetime. Spacetime is as a consequence of observation. A particle that is spontaneously decaying can experience spacetime because the decayed material become external observers.
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The reason why I start my blog with this idea is because if we consider Newton's law which can be summised as saying that...

a body in a state of motion continues in that state unless otherwise disturbed by an external influence.

Moreover, if we add to that, Einstein's idea that can be summised as...
'a body in freefall is not aware of its own motion.'

We can conclude that there is no way a moving body can tell that it is moving unless there is an external reference point. Such an object, in a sense, cannot spontaneously collapse its own wavefunction. Here, I classify spontaneous emissions (virtual particle pairs) as external observers. So that we can suggest that the Uncertainty Principle arises from the presence of external observer(s). But...why is this important?

In my opinion this implies that a freely moving particle alone in spacetime does not know that it is moving. Nor does it know that it is stationary if it cannot establish that it is moving. That is to say, according to such a particle there is no such thing as velocity. I mean that to such a particle there is no such thing as velocity(v), not even v = 0, because there is no such thing as velocity.
So, it means that velocity exists iif there is an external observer whose presence can provide a reference frame.
If we have two observers Alice(A) and Bob(B), my suggestion is that A assigns velocity to B and vice-versa.

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In the picture to the right we are shown the interaction, as it were, between the 2 observers in Lagrangian terms.

By substituting (ii) into (i) we make plain the fact that the observer is an integral part of any Lagrangian.

In this brief discussion we want to counter the argument that our Lagrangian is prone to 'instantenous action at a distance' notions by saying that we can think of such Lorentz-invariance breaking terms as geometrical artefacts rather than as actual physical transformations. Much like a gauge theory.

Lagrangian Pairs II

Now I wish to develop this particular notion of Lagrangian Pairs further.

When we build a Hamiltonian we consider the derivatives of the Lagrangian. For example, we define the momentum as shown in the picture to the right.

Lagrangian pairs.

The idea that Lagrangians come in pairs can be understood; that it is a consequence of the fact of obsever-dependent measurements.

We simply need to understand that the presence of one Lagrangian immediately implies another. We can suggest that Lagrangians come in j= 2n multiples where n takes on specific values and n>0 and j is the number of Lagrangians in the system.

There is a reasoning that we can use to come to this point of view.

Consider that from Newton...we can imagine that a particle in motion continues in its state of motion unless otherwise disturbed by an external stimulus.

From Einstein...we can understand that a particle in freefall does not sense its own motion through the gravitational field.

These facts, if we consider them in tandem, lead us to the notion that a Lagrangian cannot exist in isolation. There must always be at least two Lagrangians or more but never just one. For example, a system with three distinct energy packets has n =3/2 and j = 3. The reason for this is trivial, an isolated energy packet of any size cannot observe its own motion, there must be an observer present who can define motion. Of-course, implicit in this argument, the fact that ordinary spacetime can be defined as flat within an infinitesimally small volume or at some appropriate scale so that at such scales the background geometry does not become an 'observer'.

Notice also that the assertion that there always exists more than one Lagrangian is independent of the background geometry. However, I shall show that even though such an assertion is made independently of the background (curved spacetime or flat spacetime) it is possible to construct a geometry that identifies the same said assertion.

Waddup!

Hi,

Welcome to my blog. My name is Claver. I hope to begin and continue a dialogue in Physics and Maths with you, the global audience.

Well, my aim is two-fold: I hope that I can share my thoughts regarding aspects of Physics and Mathematics.

As well as to provide you with some insight into ideas that I hope to see added to the Standard Quantum and Classical Models of Physics.

Notice that I said added, this immediately implies that I don't fancy 'disproving' Einstein or other standard models. That's not my goal or ambition.

Hopefully, you will join me as we explore the limits of these Standard Models and build imaginative additions to them.