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Author Topic:   Missing Matter
Agobot
Member (Idle past 3789 days)
Posts: 786
Joined: 12-16-2007


Message 61 of 104 (486119)
10-16-2008 6:13 AM
Reply to: Message 59 by Straggler
10-15-2008 7:46 PM


Re: Field Theory and Particles
Straggler writes:

Apologies if the questions are dum but if they are I doubt I am alone in my lack of comprehension............

Let me add a few more questions to your confusion -

1. How can a zero-dimensional zero-volume particle have a 3D field around it that is one of the reasons we get the impression of solidness of "matter" and the non-zero volume of atoms(around it is an approximation, i am lost where in the zero the field would lie)? Does quantum physics consider such a particle to physically exist(whatever "physically exist" means outside our human perspective).

2. Would the 1 dimensional particles(strings) in String Theory resolve the above conundrum?

3. Is there anything else to our existence beside mass, energy and field?

Edited by Agobot, : No reason given.


"Science without religion is lame. Religion without science is blind"

"I am a deeply religious nonbeliever - This is a somewhat new kind of religion"

-Albert Einstein


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Agobot
Member (Idle past 3789 days)
Posts: 786
Joined: 12-16-2007


Message 62 of 104 (486169)
10-16-2008 4:18 PM


Programmable atoms
While we are on the topic of matter, i'd like to see what you think about programmable atoms. This idea has its reasons and logic, but i don't see it happening in our life-time. Basically, it says it can change atoms from one element to another with the use of quantum "dots" in place of the nucleus and changing the numbers of electrons(same thing that happens to a human body when someone dies) and for instance the technology will be able to turn water to coke.
But then that's how i view our reality and existence, if this ever becomes true we should be able to produce whatever physical reality we wish - even a small realistic "simulation" of our reality with the human beings inside it. This didn't sound right until i saw that credible universities are working on this issue. I'll stop short of saying such a technology could let us become gods, as the idea seems to be still in diapers and the authors seem to suggest the technology would not arrive earlier than the 22th century.

http://www.wired.com/wired/archive/9.10/atoms.html

http://en.wikipedia.org/wiki/Programmable_matter

http://findarticles.com/p/articles/mi_m1200/is_/ai_99187018

As tempting as it sounds, the hype does leave me with an impression of a multi-billion dollar appeal for funds for something that will possibly be intriguing in the year 2222.


"Science without religion is lame. Religion without science is blind"

"I am a deeply religious nonbeliever - This is a somewhat new kind of religion"

-Albert Einstein


    
Agobot
Member (Idle past 3789 days)
Posts: 786
Joined: 12-16-2007


Message 63 of 104 (486217)
10-17-2008 5:13 AM
Reply to: Message 59 by Straggler
10-15-2008 7:46 PM


Re: Field Theory and Particles
Straggler writes:

So a Hilbert space is a space of points where each point is a probability amplitude of all the possible field configurations.

I'd say yes, but then cavediver said QM is deterministic in this thread, so to my layman understanding it'd mean that we should be able to calculate the probability amplitudes of all points with precision and so they wouldn't really be probabilities any more. Of course, I am trying to fill my gaps of QM knowledge with reason and common sense, so i could be plain wrong on the above conclusion.

Straggler writes:

Each probability amplitude is itself infinite because the number of possible field configurations is, in practise, infinite.

I am interested to know if cavediver and san goku would go as far as to say the probability amplitudes are infinite or whether they are near-infinite(but not infinite) and locally bounded.

Straggler writes:

So what decides how many points there are in a particular Hilbert space? Is the number of points also infinite in practise? Or not? How many would there be to represent your 4 possible field states?

I'd take a wild guess and say as much as needed so that the Hilbert space could be represented and defined mathematically. But i am also interested to know if those points stand for elementary particles of the quantum field with their wavefunction in this approach of description.

Edited by Agobot, : No reason given.


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Son Goku
Member
Posts: 1150
From: Ireland
Joined: 07-16-2005
Member Rating: 7.2


Message 64 of 104 (486293)
10-18-2008 7:51 AM
Reply to: Message 59 by Straggler
10-15-2008 7:46 PM


Re: Field Theory and Particles
So a Hilbert space is a space of points where each point is a probability amplitude of all the possible field configurations.

Yes

Each probability amplitude is itself infinite because the number of possible field configurations is, in practise, infinite.

Yes

So what decides how many points there are in a particular Hilbert space? Is the number of points also infinite in practise? Or not? How many would there be to represent your 4 possible field states?

There is always an infinite number of points, even in my simpler example of only four possible states. Basically this is because any amplitude can be written as:
(a,b,c,d) Where a,b,c and d are the probability amplitudes for each field configuration. a,b,c and d can be any real number and since there are an infinity of such numbers there is an infinity of lists.

How many dimensions does a given Hilbert space have, what decides this, what do these dimensions physically reperesent?

Basically the number of dimensions of a Hilbert space is equal to the number of classical states. A classical bit of information can be either 0 or 1, but a quantum bit of information (a qubit) is a list of probability amplitudes for being in either 0 or 1. Such as (a,b). Where a is the probability amplitude for 0 and b is the probability amplitude for 1. So as you can see the amount of numbers in the list is basically dictated by the number of classical states.
For a field or even a normal particle, there is an infinite amount of states. Hence an infinite number of dimensions.

The only slight qualifier is for properties like energy. To take the example of a hydrogen atom, classically there can be any number of energy states. An electron can have any energy it wants in the classical theory.

In the quantum theory however one finds that the lists of probability amplitudes only list the probabilities for states of certain energies, some classically allowed energy states have no probability amplitude and hence the system has no chance of possessing that energy, an energy that was allowed in the classical theory.
For a hydrogen atom the fact that only certain energies can be obtained is what actually prevents an atom from collapsing.

However roughly:
dimension of Hilbert space = number of classical states.

Apologies if the questions are dum

They certainly aren't.
This message is a reply to:
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Straggler
Member
Posts: 10285
From: London England
Joined: 09-30-2006


Message 65 of 104 (486442)
10-20-2008 4:22 PM
Reply to: Message 56 by Son Goku
10-12-2008 2:35 PM


Re: Field Theory and Particles
Also since the original field configuration probability amplitudes obeyed relativity and causality, the new particle ones will as well. Voila!, a theory of relativistic quantum particles.

QM is often cited as being non-causal in the sense that, for example, we cannot predict which atoms will decay we can only say how many atoms will decay. In this sense QM is considered to be inherently probabalistic and "non-causal".

Does QFT give rise to causality in this sense or am I getting my "causalities" mixed up? What exactly did you mean by causality in this context?


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Straggler
Member
Posts: 10285
From: London England
Joined: 09-30-2006


Message 66 of 104 (486444)
10-20-2008 4:30 PM
Reply to: Message 64 by Son Goku
10-18-2008 7:51 AM


Re: Field Theory and Particles
If we considered a simple example such as a single electron and the corresponding electric field how would we go about constructing our probability amplitudes and constructing our Hilbert space?

Or is this where things get impossible without the required maths?

Even if the process of constructing the Hilbert space for such an example is beyond this discussion could you describe what the Hilbert space for a such an example would look like?


This message is a reply to:
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Son Goku
Member
Posts: 1150
From: Ireland
Joined: 07-16-2005
Member Rating: 7.2


Message 67 of 104 (491621)
12-18-2008 12:03 PM
Reply to: Message 65 by Straggler
10-20-2008 4:22 PM


Re: Field Theory and Particles
Apologies for the enormous delay. I was busy with research, but I've given some thought to your questions and believe my answers will be superior to what they would have been if I'd responded initially.

QM is often cited as being non-causal in the sense that, for example, we cannot predict which atoms will decay we can only say how many atoms will decay. In this sense QM is considered to be inherently probabalistic and "non-causal".

Does QFT give rise to causality in this sense or am I getting my "causalities" mixed up? What exactly did you mean by causality in this context?


Causal, to a physicist, is meant in the sense of special relativity. It means that the influence of one physical system on another cannot travel faster than the speed of light.
Quantum Mechanics is non-deterministic or statistical. Everybody knows what this means so I won't explain it.

Quantum Field Theory is still probabilistic or non-deterministic (which ever word you want), however it is causal, in that nothing travels faster than light.

The reason the word causal is associated with staying below the speed of light is because if the speed of light could be exceeded then the world would have no real past--->future flow of time that anybody could roughly agree on and would lead to time travel.

QM might have random events but they are nicely ordered in a temporal sense, if you get my meaning, e.g. This atom randomly decayed before this other atom. So QM is causal. Or rather QFT is causal as standard QM doesn't have relativity built in.


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Son Goku
Member
Posts: 1150
From: Ireland
Joined: 07-16-2005
Member Rating: 7.2


Message 68 of 104 (491622)
12-18-2008 12:21 PM
Reply to: Message 66 by Straggler
10-20-2008 4:30 PM


Quantum Probability 1
If we considered a simple example such as a single electron and the corresponding electric field how would we go about constructing our probability amplitudes and constructing our Hilbert space?

Or is this where things get impossible without the required maths?

Even if the process of constructing the Hilbert space for such an example is beyond this discussion could you describe what the Hilbert space for a such an example would look like?


I've come up with what I think is a good example of this stuff. It'll take a few posts. I'm going to exploit the fact that Quantum Mechanics, in mathematical sense, is more general version of a branch of maths everybody is familiar with. Probability.

First I'll start with something simple, the Hilbert space of a classical coin toss.

Let me take a fair coin, one that is equally likely to land heads or tails. If I have such a coin then I can measure several observables as physicists call them. For example if +1 is heads and -1 tails I could measure the average of all coin tosses. I could measure the average of the cube of each coin toss, e.t.c. Of course for a coin there aren't really that many interesting observables. Any way for each observable I'll get numbers for all my averages. From this list of averages I can figure out all statistical information about the coin that I could possibly want.

If I use an unfair coin, one with a bias to heads for instance, then I'll get a different list of averages which would lead to different statistical information. This is to be expected since an unfair coin should be different statistically from a fair coin.

Now for every single way the coin could be unfair (and the the one case when it is fair), that is for every different possible state of the coin, I'll get a different list of averages.

So I can label each state of the coin by its list of averages. The collection of these lists of averages is then called the Hilbert Space of the coin. Since each list of averages belongs to one state of the coin, this Hilbert space is also known as a state space.

If this makes sense I'll move on to QM (which is actually a very simple generalisation of all this) and then entanglement will be quite easy to explain. If fact spin and entanglement are actually just features of the quantum analogue of a coin toss.

Edited by Son Goku, : Better title.


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Stile
Member
Posts: 3846
From: Ontario, Canada
Joined: 12-02-2004
Member Rating: 1.7


Message 69 of 104 (491626)
12-18-2008 1:39 PM
Reply to: Message 68 by Son Goku
12-18-2008 12:21 PM


Re: Quantum Probability 1
Son Goku writes:

So I can label each state of the coin by its list of averages. The collection of these lists of averages is then called the Hilbert Space of the coin.

That is to say:

The Hilbert Space of a coin is the list of all possible averages that any coin can have.

Would dimes have a different Hilbert space than quarters? Or is a Hilbert space powerful enough to describe all "two-sided coins" or something? Or is this not a good question for this kind of analogy?

I assume a Hilbert space includes all theoretical coins? That is, it's quite possible to have a state in the Hilbert space for a coin that does not actually exist, correct?


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 Message 68 by Son Goku, posted 12-18-2008 12:21 PM Son Goku has responded

Replies to this message:
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Stile
Member
Posts: 3846
From: Ontario, Canada
Joined: 12-02-2004
Member Rating: 1.7


Message 70 of 104 (491627)
12-18-2008 1:41 PM
Reply to: Message 68 by Son Goku
12-18-2008 12:21 PM


Re: Quantum Probability 1
Double Post. Whoops.

Edited by Stile, : My first double post! Yay??


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Son Goku
Member
Posts: 1150
From: Ireland
Joined: 07-16-2005
Member Rating: 7.2


Message 71 of 104 (491679)
12-19-2008 11:35 AM
Reply to: Message 69 by Stile
12-18-2008 1:39 PM


Re: Quantum Probability 1
The Hilbert Space of a coin is the list of all possible averages that any coin can have.

Yes, although I'll say it a different way for clarity.

There are several different quantities or observables related to a coin that can have averages.

To illustrate, consider the following list:
(Average of Coin Toss, Average of double coin toss, Average of thrice of coin toss, ......)
(+1 is heads, -1 is tails.)

For a fair coin I get:
(0,0,0,......)

For a biased coin I might get:
(0.66, 1.5, 2.25,.....)

The first list is a list of averages for a fair coin, the second a list of averages for a biased coin. However it is the collection of all possible such lists that is the Hilbert space.

So the Hilbert Space is a "space" of lists and any given list is a list of all the averages of different quantities.

Would dimes have a different Hilbert space than quarters? Or is a Hilbert space powerful enough to describe all "two-sided coins" or something? Or is this not a good question for this kind of analogy?

Well, the funny thing is, this isn't really an analogy. The above description is actually the Hilbert space of a coin toss.
As for if it would be different for a dime or a quarter, well two things. First, I'm not that familiar with what a dime and quarter are actually like. Looking at wikipedia they seem to be roughly similar circular coins. Second, the differences between them come down to things like composition and mass, e.t.c. The way I'm discussed things above, I didn't really look at these kinds of differences. However it doesn't matter, for if I was to take these things into account, dimes and quarters would behave like one of the biased lists, a different list for each. (Both lists would be very close to the fair coin list, since currency coinage is quite fair when it comes to a toss). However a list of averages is still a list and so it is in the Hilbert space. So you're correct with "the Hilbert space is powerful enough to describe all two sided coins". All you do is choose the right list.
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shalamabobbi
Member (Idle past 1108 days)
Posts: 397
Joined: 01-10-2009


Message 72 of 104 (494410)
01-15-2009 7:52 PM


bump
Bumping hoping to revive this thread..
I went to school before quarks and dark matter were discovered.
I have no cosmological background.

Some of the theorized candidate particles for dark matter exist at higher energy levels than can be detected by existing colliders

Dark Matter interacts either not at all or very weakly with other matter through electromagnetic, strong or weak forces.

One way of telling would be if the particle that was created was very stable, since Dark Matter doesn't seem to decay.

Do any existing theories predict these particles, or are they expected only from the observations of galaxy rotations?

How does dark matter affect the BB theory? I thought it was particularly sensitive to initial conditions.

Why would it distribute itself differently from visible matter since gravitationally it is equivalent to visible matter?
(Why wouldn't some of it reside in our solar system and be detectable as gravitational anomalies?)

The simple answer is that when you formulate string theory in d space-time dimensions, you obtain an anomaly which destroys the physical characteristics of the theory - however the anomaly contains a factor of (d-10), so when d=10, the anomaly disappears.

Does it disappear for a number of higher dimensions as well, or just for 10?

Are the extra dimensions real(behind the scenes) in the sense that only solutions that end up in our space-time work? Or is there a prediction of some sort of hidden reality that doesn't need to manifest itself in our space-time. (can something real hide in the 6=10-space-time dimensions, and exist apart from our knowledge of it?)
My understanding is that these extra dimensions are not extensive like those of 3D space, is that correct?

(If anybody is familiar with linear algebra, a Hilbert space is just a type of vector space and rewriting things in terms of particle quantities is just choosing another basis for the vector space.)

Does that imply that the basis representing particles is real whereas a different basis would not be real although mathematically possible? Are particles real or does the concept/basis simply make the calculations more tractable?
Does this have anything to do with string theory's 10 dimensions? Are these dimensions necessary to create particles in space-time?

Edited by shalamabobbi, : No reason given.


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Son Goku
Member
Posts: 1150
From: Ireland
Joined: 07-16-2005
Member Rating: 7.2


Message 73 of 104 (501620)
03-07-2009 5:14 AM
Reply to: Message 72 by shalamabobbi
01-15-2009 7:52 PM


Re: bump
My apologies for being so late to reply to this.

Do any existing theories predict these particles, or are they expected only from the observations of galaxy rotations?

Several theories predict such objects. In fact very simple extensions of the standard model of particle physics include such objects.

How does dark matter affect the BB theory? I thought it was particularly sensitive to initial conditions.

Good observation. Dark Matter doesn't really change the Big Bang theory because adding it in doesn't really change which solution to Einstein's Field Equations describe the universe.
Okay technically it does change it, but it's still a Big Bang spacetime with all the same features, it only effects stuff such a galaxy rotation.

Why would it distribute itself differently from visible matter since gravitationally it is equivalent to visible matter?
(Why wouldn't some of it reside in our solar system and be detectable as gravitational anomalies?)

There studies being started which will attempt to get some grip on how Dark Matter is distributed. Dark Matter is still clumpy like ordinary matter as it seems to surround normal matter galaxies. There would be some in our solar system, but not enough to really do anything.

My understanding is that these extra dimensions are not extensive like those of 3D space, is that correct?

Yes, the extra dimensions in String Theory are small and curled up on themselves.

Does that imply that the basis representing particles is real whereas a different basis would not be real although mathematically possible? Are particles real or does the concept/basis simply make the calculations more tractable?

The particles are simply more mathematically tractable. The objects described by another basis would be just as real. However I'll explain in my next post why we use particles.
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 Message 72 by shalamabobbi, posted 01-15-2009 7:52 PM shalamabobbi has responded

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Son Goku
Member
Posts: 1150
From: Ireland
Joined: 07-16-2005
Member Rating: 7.2


Message 74 of 104 (501622)
03-07-2009 5:59 AM
Reply to: Message 68 by Son Goku
12-18-2008 12:21 PM


Quantum Probability 2
My extremely slowly delivered series on Quantum Field Theory continues...

So I've just described the Hilbert Space for a coin. The important points:
1. The list of all averages for a coin (see above) is called a state since it describes the coin.
2. The Hilbert space is the set of all such states, the set of all possible coins.

Now obviously, related to what Stile said above, if two coins are identical statistically, or the exact same as far as a gambler is concerned, then they are identical in this Hilbert Space way of viewing things. So a Hilbert space doesn't really describe coins completely. However it does describe particles completely.

Quantum Particles:
In Quantum Mechanics all have to do to my list of averages is make it noncommutative. What this means is actually quite simple to understand and is the cause of the uncertainty principle.

Let's go back to the old classical example of a roulette wheel. Just like the coin I can make a Hilbert Space description of the Roulette Wheel. Let's say I'm working with a Roulette Wheel where you use two balls on every spin and one has to land on red and another has to land on black. Then one average, in my list of averages, is the average value of the black number multiplied by the red number. I'll call this quantity:
B.R

Now obviously this is the exact same as the average value of the red number multiplied by the black number:
R.B

So, B.R = R.B

So in my list of averages I'll get two quantities which are the same, for example:
(..., 34, ......., 34,.......)
All quantities which are just things multiplied in a different order will have the same average.

Now let's turn to a quantum particle. Two things we can measure are position, X, and momentum, P. Just like the Roulette wheel and the coin toss, I can make a list of averages. However position times momentum:
X.P

Will not have the same average as momentum times position:
P.X

Quantum Mechanics is probability where the order of multiplication matters, if the order of multiplication matters in mathematics we say it is noncommutative.

Now in an experiment in order to measure X.P you would measure position and right down the result, then measure momentum and right down the result and then multiply them.

To measure P.X you do the same thing, but measure momentum first. However because order matters, this tells you that as you measure X.P and your colleague measures P.X over and over again, the average result in the two cases will differ. Measuring momentum first is different from measuring position first.

So the position of a particle is affected by measuring momentum before it. If we already knew the position of the particle before hand and then measured momentum and then measured position again, the second position measurement would be altered by the fact that a momentum measurement proceeded it. So you will get a different answer from the first time you measured the position, hence measuring momentum ruins your previous knowledge of position. It works the same with momentum. Measuring one ruins your knowledge of the other, hence the necessary uncertainty in position and momentum.

All because nature cares about which way things are multiplied.

Was this helpful?


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Son Goku
Member
Posts: 1150
From: Ireland
Joined: 07-16-2005
Member Rating: 7.2


Message 75 of 104 (501644)
03-07-2009 9:14 AM


Quantum Probability 3
I will now begin to add some bits and pieces that are necessary before I begin to introduce relativity into quantum mechanics.

Thus far I have said that quantum mechanics is described by a Hilbert Space. The Hilbert space is a set of states and each state is a list of averages. What distinguishes quantum mechanics from regular probability is that these averages have different values for quantities which are different orderings of the same thing, like X.P and P.X

Time Evolution:
The next important thing is time evolution. Obviously things evolve in time in the real world. This is also true in quantum mechanics. It occurs by the list of averages changing over time. So lets say I watch a system over a time period of three seconds. I could get the following (t is the time in seconds):
(0.4, 0.5, 0.01,.......) t = 1
(0.2, 0.8, 0.12,.......) t = 2
(0.3, 0.1, 0.94,.......) t = 3

So the values of the averages are changing over time.
Since a "state" is a list of averages and the particle is changing averages, it is changing its state over time. Its moving from one state to another in the Hilbert space.

How it changes its values is described by the Schrödinger equation, the central equation of quantum mechanics.

Two or more particles:
If I have two particles the Hilbert space gets a little more involved, but not greatly so.
Basically the lists will be a lot longer because you have:
1. The averages of the every quantity to do with the first particle
2. The averages of the every quantity to do with the second particle
3. The averages of the every quantity that involves both particles, for example their total spin or total energy.

For three particles one has:
1. Averages associated with one particle
2. Averages associated with any pair
3. Averages associated with all the particles

And so on if you have four, five, e.t.c.

In the two particle case, lets say I have a list:
(...1...;...2...;...3...)
1,2,3 mean the different types of averages I mentioned earlier.

Sometimes these two particle lists can be broken done into two separate one particle lists, so the particles can be viewed and understood separately.

However some two particle lists can't be broken up like this, the particles must have a single combined list, they cannot be understood separately. This is entanglement.

Now some notation for the next post. If I write a list like this:
{2}(.........)
I mean a two particle list.
{5}(.........)
is a five particle list and so on.

The next post introduces special relativity and will explain why we need fields and not just particles.


  
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