Fields Classical and Quantum

[studiot]

This thread was inspired by a question asked in a thread about stored energy where the question was asked if energy could be stored in a Field.
Since this was really a bit off topic in that thread and an important subject people are always asking about this and the nature of Fields in general here is a thread for folks to discuss the subject and ask their questions.

For reference I described a Field this way in that thread.

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So to start with a region of space.
This is pretty much the same as its English counterpart and might be the inside of an eggshell or a smartie box.
More usefully in Science it might be the bore of a pipe or the interior of a chemical reaction flask.
Or of course it could be the whole of space itself or just the space around planet where the panetary gravity is not negligable.

Whatever region we take it, we need a boundary to describe what lies inside the region and what lies outside.
Proper specification of that boundary is of vital importance in Science, particularly Thermodynamics - the Science of Heat.
We also identify  every individual location or point within the region. Some of these have special names such as the centre of gravity.

Another English word with a special meaning in Science is the word Field.

In English it often means 'the stage in which the activity takes place' eg the 'field of operations', 'he was an expert in his field' and so on.

This meaning is not taken on by Science, so do not use it in a scientific manner.

In Science a Field is a region of space where every point of that region has a specific value of some property of interest assigned to it.
Every point might have the same value eg the density in a homogeneous substance, or the pressure inside a small balloon.

So we can talk of a density field or a pressure field
Or the value may be different at every point for example the temperature field within a large copper bar, heated at one end with a blowtorch.

The properties of that Field as a whole will depend upon the particular property we are considering.
Some of these properties as simple numbers like in my examples, some are much more complicated mathematical expressions, such as the electric field vector in an electric field or the magnetic field vector in a magnetic field.

But there may be many properties and other things (like machinery or electric fires) inside a region.
In themodynamics we call such a region a 'system' and we include, but separately identify, the boundary.

These concepts allow us to develop conservation laws.

Conservation of mass, conservation of momentum, conservation of energy.

 

So the quick answer to the first question is yes if the field property is suitable, no if the field property is not.

In order to illustrate this and another important question - What are the differences and similarities between classical and quantum fields here is a comparison.

 

Consisder a stream running in its channel.

At every point in the stream we can assign (and even observe) a velocity vector.

The region is obviously the stream channel and chosen fields could be

A) The magnitude of the velocity.

B) Just the direction of the velocity.

Both Fields are purely classical.

Field (A) can be used to calculate the kinetic energy of the water, if we know its density.
So Field (A) can be used to describe energy and indeed if the water is halted this energy is transferred elsewhere.
For instance if the water encounters a dam and builds up, the up implies an increase in potential energy, which is exactly what happens.

This energy is 'stored'.

TO BE CONTINUED.

 

Edited September 30, 2017 by studiot

[studiot]

CONTINUED

Field (B) - the direction field - however cannot be used to store energy, as energy is not a viable function of direction.

 

Now the statement  is often made 'A particle is just a disturbance in the Field' and often accompanied by some wooly waving which means 'but I'm not sure how'.

So let us use these fluid examples to display classical disturbances that can be localised and therefore have some particle like properties.

Suppose there are some submerged rocks in the stream.

The stream fields will change in response. Whirlpools will develop in the vicinity of the rocks.

The direction field will show the whirlpools, the magnitude field will show something else.

In the whirlpools the velocity will diminish and, as we have seen, some kinetic energy will be transferred to potential energy.

Eventually a fairly stable flow regime will establish itself where the whirlpools represent 'particles' which for instance deflect passing small objects floating in the water, just like forces between massive bodies.

 

OK so classical Physics an develop particle like entities in its Fields, what about Quantum Physics?

 

The Classical Fields developed its 'particles' by interaction between the Field and its environment.

This also happens in the Quantum Field.
The variable property in the Quantum Field is the Wave Function.
This is quite different from the classical point functions we saw in the stream.
Without interaction it is non localised - that is it extends in theory over all space, although it is usual to restrict it to that region where the contribution is significant.

 

[Mordred]

I like your use of the stream example, it highlights an example of a charged field as opposed to a scalar field. I plan to expand on some of the details you raised in the above. 

 Once RL allows proper focus time lol. The above will make a good example to explain the stress tensor elements. Hence wanting time to concentrate.

Edited October 1, 2017 by Mordred

[Mordred]

On 30/09/2017 at 11:26 AM, studiot said:

 

At every point in the stream we can assign (and even observe) a velocity vector.

The region is obviously the stream channel and chosen fields could be

A) The magnitude of the velocity.

B) Just the direction of the velocity.

Both Fields are purely classical.

 

Excellent summary Studiot I would just like to add some essential details without specifying any particular theory.

a) Scalar field (magnitude only), there can be tons of smaller movements, but were concerned with the global average mean value. This would be a spin zero, field of value x. There is no inherent direction component of this field. Using relativity as an example the vacuum solutions is one example. In the FRW metric inflation. This is a homogeneous and isotropic field.

b) The field now has a direction component mean average, this is no longer a spin zero statistic but the field spin will depend on the quage bosons involved. For the electromagnetic field, spin 1. Gravity best matches spin 2. Any field with an inherent direction is a charged field. Spin will depend on the charges involved.

(treat charge as simply attraction/repulsion).

 

there is a handy formula that applies to demonstrate the second post by Studiot. Scalar field equation of state.

[latex]w=\frac{\frac{1}{2}\dot{\phi}^2-V\phi}{\frac{1}{2}\dot{\phi}^2+V\phi}[/latex] the numerator is the kinetic energy the denominator is the potential energy.

A field with a direction would need additional terms to that equation. Terms mentioned in Studiots last post of flux and vorticity.

 

 

Edited October 3, 2017 by Mordred

[MarkE]

Thanks @studiot. When, in your example, "the kinetic energy will be transferred to potential energy", where did the energy exactly go? Is it still measurable, but in another field? 

By the way, thanks for your recommendation of that book on thermodynamics. Next on my schedule is the TTC lecture on thermodynamics, so we'll see about that (if I still have questions afterwards)

[studiot]

22 minutes ago, MarkE said:

Thanks @studiot. When, in your example, "the kinetic energy will be transferred to potential energy", where did the energy exactly go? Is it still measurable, but in another field? 

By the way, thanks for your recommendation of that book on thermodynamics. Next on my schedule is the TTC lecture on thermodynamics, so we'll see about that (if I still have questions afterwards)

Hello again Mark, nice to see you made your way here.

Yes, the energy went into another field. I hiope you did not think that my direction and magnitude fields were the only possible ones we could choose for the stream.

There are in fact a large number of possibilities, each with its own charateristic and use.

The other field is the fluid pressure field, which I why I called it potential energy.

The pressure in a fluid is also a function of the magnitude of the velocity, just as is the kinetic energy.

There is a theorem, called Bernoulli's Theorem, that connects these energies by way of an equation.

The greater the magnitude of the velocity the greater the kinetic energy, but the lower the pressure (energy).

Conversely the lower the velocity the greater the pressue and the lower the kinetic energy, until when the fluid is standing still or stagnant there KE is zero and the energy is all pressure (which is at its greatest).

Both these forms of energy are capable of bineg transferred to another body outside the fluid, as work.
If that happened then that energy would be lost to the fluid.

[Dubbelosix]

Hey, you should write up about how the vacuum relates to idea's about the classical spacetime and the quantum spacetime. For instance, how does the classical spacetime diverge from the quantum? One feature is the presence of vacuum fluctuations. It's nice to see someone creative, even though I haven't read it all. 

The classical vacuum is also known as a Newtonian vacuum and differs because in the ground state a Newtonian vacuum is a pure vacuum with a zero expectation value - while in quantum mechanics, it is not zero!

Edited October 10, 2017 by Dubbelosix

[studiot]

On 10/10/2017 at 7:58 PM, Dubbelosix said:

Hey, you should write up about how the vacuum relates to idea's about the classical spacetime and the quantum spacetime. For instance, how does the classical spacetime diverge from the quantum? One feature is the presence of vacuum fluctuations. It's nice to see someone creative, even though I haven't read it all. 

The classical vacuum is also known as a Newtonian vacuum and differs because in the ground state a Newtonian vacuum is a pure vacuum with a zero expectation value - while in quantum mechanics, it is not zero!

WEll thank you for the words of support, Doubbelosix. I is nice to have a new member who goes around throwing out words of common sense rather than fanciful assertions that cannot be substantiated.

 

You want to talk about space or spacetime?

Perhaps we should develop that interesting issue in another thread?
I looked at your thread about this but it was all cosmological, which is not my bag really.
So if you like we can start a new thread about it.

I have said here that Fields belong in some particular region of space, but I haven't specified what sort of space, whether it is volumetric space or phase space for instance.
The interesting thing about Fields is that if the field property in n-space is a scalar, we can drop perpendiculars to the base to create surfaces or hypersurfaces with height equal to the scalar value of that propery.

The whole of shooting match exists solely in the volumetric space of n+1 dimensions, although it is difficult to imagine the 4D version needed for our 3D world.

But if the Field is a vector or tensor we cannot do this.
The vectors and tensors exist in a space of their own which touches our volumetric space at a single point.
This is often overlooked when drawing vector arrows, but is easy to demonstrate.

[Mordred]

This isn't complete yet, I have been working on it for roughly a week now off and on as RL allows. As this thread is appropriate and I'm positive Studiot won't mind. (Particular I was trying to formulate the vector fields as simply as possible).

I figured I would post it here to gather recommendations comments etc and tie in with Studiots engineering field treatments. Quite frankly the mathematical formulations at their rudimentary physics are essentially identical. The deviations arise in application 

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One of the topics of interest is the study of field treatments in physics. This study encompasses a vast array of different treatments under a plethora of different metrics. A layperson trying to learn how to model fields is faced with a daunting challenge if doing so as a self taught study. Even under formal educational systems, the topic itself presents challenges in terms of its diverse nature of applicable mathematics and potential treatments.  My goal as per a request by another forum member is to help clarify what fields are, why they are so important in physics and detail some of the fundamentals to basic field modelling. In essence detailing the essentials of fields that is inherent in any field treatments.

 So to begin with lets start with the primary question of interest. What is a field?, and the related oft asked question "what makes up a field"?

A field is an abstract tool under a coordinate basis. where each coordinate are assigned one or more values. (All mathematical treatments incuding physics is a representation.) Those representations have boundaries where the representation applies. Care must be taken in remembering this when studying any physics topic.

Essentially it is a collection of objects, events, values of any kind applied to a geometry or metric for short. So the second question of "What is it made up" ? is already addressed by the simple fact, it is an abstract device, that we can assign any function under a given arbitrary metric treatment. So in Cartesian coordinates [latex]\mathcal{F}(x,y,z)[/latex]. I will denote field with [latex]\mathcal{F}[/latex].  Under polar coordinates [latex]\mathcal{F}(r\theta\phi)[/latex]

Coordinate systems : As mentioned above, we map our quantities under a coordinate basis, however we place no priority on any particular coordinate system. Why should Euclidean flat coordinates have a higher priority than Cartesian or cylindrical ? or The Schwartzchild vs Newton or any other coordinate system under QFT or String theory ?. They all have their range of validity and accuracy dependent upon the objects/system under examination. The Newton approximation is equally valid in our everyday lives. It is only when relativistic effects become measurable under examinations, that we require GR. The aforementioned theories each has its strengths and weaknesses when compared to one another.  As the expression goes, "The universe cares not how we measure it". Every theory evolves as new research is presented, so one should be aware that new developments may or may not address previous problems within a given theory.  However there is some basic geometries common to all physics theories. Though they may be represented under lower and higher dimensions. (see degrees of freedom). 

The most commonly known is the Euclidean geometry that we are all familiar with.

-Euclidean: Everyone is familiar with the 3 dimensions, (x,y,z). These are orthonormal axis, they are 90 degrees from one another. "In linear algebra, two vectors in an inner product space are orthonormal, if they are orthogonal and unit vectors. A set of vectors form an orthonormal set, if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis." https://en.wikipedia.org/wiki/Orthonormality.

also see list of trigonometric identities, as these are preserved in the Euclidean frame.

https://en.wikipedia.org/wiki/List_of_trigonometric_identities

it is only under relativistic affects do we require transformations to preserve the trigonometric identities in particular Pythagorous theorem.

[latex] c^2=a^2+b^2[/latex]

in 3d a common one can apply to the last

[latex]s^2=x^2+y^2+z^2[/latex] for the purpose of this article I will denote this with the lower case for S, to avoid confusion with upper case S for separation. In the former s is the longest line of a 3 dimensional triangle.

In the Euclidean frame there exists a theory of relativity called Galilean relativity and subsequently Galilean invariance. Although this article is not specifically about GR and SR, All modern treatments employ the tenets of GR. So we will detail the transformation rules between absolute time and variable time in accordance to observers and some common different classes of observers. In order to model time we add time as a fourth dimension. A dimension under physics and mathematics is any independent variable. So four dimensions 3 spatial one of time. (t,x,y,z) to give time dimensionality of length, we apply the constant c, not to be confused with the speed of light. (ct,x,y,z) so a field can now be represented by [latex]\mathcal{F}(ct,x,y,z)[/latex]. In polar coordinates. [latex](ct,r\theta\phi)[/latex]. (surface of a sphere such as the Earth.)

Degrees of freedom and dimensions:  A common confusion arises, by a vast majority of laymen first learning any field treatment. That being the use of higher dimensions beyond the 3 spatial and 1 time dimension. One has to understand a dimension under physics  is an independent variable. This is any function that can change without affecting any other function. These include each spatial dimension and under GR the dimension of time. However one can add other variable to describe how an object evolves. Under symmetry we can classify how systems/objects/states/fields evolve under translations under three distinct categories. Spatial, rotational and time translations. Take for example a rotor blade, if we move the blade from one location to another without changing its orientation, the object (blade) does not change only its location. This is a spatial translation. Now take that same blade and change its orientation to any of the principle axis. This is a rotational translation. Translations in time are  rotational translations however under time transformations between reference frames. When [latex]\acute{t}\neq t[/latex]. One can arbitrarily break any object into smaller slices and apply a geometry and translations to those smaller slices. For example a 3 dimensional volume can be broken down into two planes. [latex]\mathcal{R}^3=\mathcal{R}^2\otimes\mathcal{R}^2[/latex]. These treatments add variables that can independently change. Which is an added dimension. For example in Kaluzu-Klein we have the 4D coordinates (3 spatial, 1 time) then we add charge with two polarities. We can describe charge via a binary function, which becomes a degree of freedom. We now went from 4d to 5d.

 

Scalar Fields :     A scalar quantity is a quantity that is not a vector, a vector is composed of a magnitude and direction. A scalar is a quantity that can be described by a single real number.

https://www.encyclopediaofmath.org/index.php/Scalaral

also see definition of quantity under mathematics.

https://www.encyclopediaofmath.org/index.php/Quantity

 Recall earlier that a field can be assigned any function, one can assign any scalar function however complex provided its resultant returns a scalar quantity. One such example is the gradient of a scalar field such as used in temperature of the Earths atmosphere. Scalar fields can also be used with curvature, an example being the temperature or pressure scalar mappings of the Earths atmosphere. The function assigned to all coordinates to the metric mapping return a scalar quantity. Prime example being [latex]\rho[/latex] for energy/mass density.

Vector Fields

diverging curl

converging divident

rotating curl

 

Galilean relativity.

[latex](\acute{t}=t), (\acute{x}=x-vt), (\acute{y}=y),(\acute{z}=z)[/latex]

Last few lines are just topics to cover, don't associate curl div, grad to the diverging converging etc. I just placed key words down to cover.

 

 

 One of the challenges above is detailing a zero'th order function (scalar), First order function ( vector) etc and keeping the math low enough for a low grade level high school student as a possible reader while still having enough detail to interest the older crowd like me

@Studiot feel like providing a real world example of contour map, with image of course including the gradiant in field treatment. ? I welcome contributions of others in the completion of the above. Obviously credits will be applied for any volunteer efforts.

(No interest in publishing, this is for the Forum usage and I might include on my website, see signature.)

Edited October 12, 2017 by Mordred

[Dubbelosix]

14 hours ago, studiot said:

 

 

You want to talk about space or spacetime?

sure