# Are any two electrons equal mathematically ?

1. Nov 28, 2005

### nameta9

In various threads dealing with why mathematics is so good at describing physics, the fact that elementary particles are exactly equivalent is really weird. Are any 2 electrons exactly equal in all possible mathematical sense ? Is this really conceivable and possible ? I find it hard to imagine 2 electrons that are exactly equal as if they were 2 exactly equal numbers. So electrons would end up being pure mathematical objects. They would in essence be pure abstractions. Maybe quantum indetermination and virtual particles and feynman diagrams of electrons self interacting make it so that NO TWO ELECTRONS ARE REALLY EVER EXACTLY EQUAL.... to any decimal place. Very odd problem indeed.

2. Nov 28, 2005

### oldtobor

No 2 electrons are exactly equal because they have different positions in space and undergo slightly different influences by their surrounding environment. So they have slightly different coordinates, speed, direction etc. Also they are surrounded by a unique and ever changing cloud of virtual particles popping up out of nowhere and disappearing constantly changing the energy configuration surrounding the electron as compared to another. If you could see these on a graph they would be very different. A better question is if the law of conservation of Energy is an approximation or exact to an infinte degree (or infinite number of decimal points). Quantum principles say that :

(E1-E0)X(T1-T0)=h (planck) means that any energy even extremely high can pop up out of nowhere as long as it is for a short enough time. I wonder if some energy can simply disappear and appear meaning that Energy conservation is not an exact absolute law. Some particle reactions may simply loose some energy or gain some without any reason at all because the law of conservation of energy is just an approximation. This would solve the riddle of dark matter and dark energy.

If the law of conservation of Energy was exact to any decimal number and absolutely true, this would mean that physics-matter and the universe are truly a mathematical abstraction, a pure mathematical / conceptual object. This would be very strange indeed. The law of conservation of energy must break down somewhere in some way, it can't be infinitely precise.

Last edited: Nov 28, 2005
3. Nov 28, 2005

### Tom Mattson

Staff Emeritus
Not only do I find it hard to imagine this, I find it downright impossible. Electrons aren't numbers, and it makes no sense whatsoever to say that two electrons are "equal".

4. Nov 28, 2005

### nameta9

If you know quantum theory, well the theory operates on and is experimentally demonstrated that elementary particles like electrons are exactly equal in all senses as they are completely interchangeable and indistinguishable. All chemistry works on this and there is no way to distinguish 2 hydrogen atoms or 2 atoms of any element except for their coordinates (and even for the positions of particles there are strange paradoxes). They have no "individual" identity, they are absolutely and mathematically identical. It is this aspect that is so striking as these elementary objects - particles - atoms etc. are TRULY PURE MATHEMATICAL OBJECTS!

An electron is defined only according to its mathematics. Think about this carefully and you will realize how insane it really is!

Last edited: Nov 28, 2005
5. Nov 28, 2005

Staff Emeritus
Not so. The electron was discovered by J.J. Thomson in 1899, long before the quantum math was developed. He determined it was a particle by the constancy of its e/m ratio on different cathode ray paths.

The quantum math is a device for getting numerical answers, not for defining.

6. Nov 28, 2005

### Tom Mattson

Staff Emeritus
I already addressed this point of yours at SFN, as have several other people.

In the thread QED: science meets science fiction

nameta9: Since there are an infinite number of diagrams according to an infinite number of possible interactions and decay modes (electron emits virtual photon that becomes virtual e+e- pair etc.) then matter is truly reduced to pure mathematics.

swansont: The descriptions are mathematical. That doesn't mean the phenomena described are nonexistant.

Tom Mattson: ...Because a mathematical description of a physical law is just that: a description. The fact that it is mathematical does not in any way rob electrons of their physical reality.

The thing that you consistently fail to realize is that while your view that physical objects really are mathematical objects is consistent with what we know, it is not in any way implied by it.

As you've been told repeatedly by different people in different threads and at different forums, this is simply not true.

I agree that it is nuts to consider that physical objects are merely mathematical objects. So given that not one iota of our scientific knowledge even remotely implies that such is the case, I cannot figure out why you keep hammering on this point.

Last edited: Nov 28, 2005
7. Nov 29, 2005

### oldtobor

In the macroscopic world it is impossible to find any 2 objects that are exactly the same as are elementary particles. I think this is the aspect that is being addressed. You may have any 2 objects, 2 pentiums or 2 car tires or even 2 bacteria and they will be different in the sense that there are some imperfections or details that make them different. It seems natural that it be so. But when you go down to the "microscopic" level say already at molecules, things become exactly the same. You can find 2 molecules that are exactly the same, the individual imperfections of the macroscopic world that identify macroscopic objects uniquely disappear. In this somewhat "aesthetic-metaphysical" sense, elementary particles seem to coincide with pure mathematical entities.

Actually come to think about it, particles such as an electrons or even more so photons have alot of metaphysical properties in common with numbers. They are both universal, they are interchangeable, they are indistinguishable, they are everywhere. So maybe we should add some physical properties to numbers and devise a new corresponding mathematics, like a number shall also have coordinates, a spin, an angular momentum etc. That would be an interesting metaphysical theory. And adding properties to numbers, you could add an infinite amount of new properties and devise a mathematics with all kinds of new constraints and results (maybe call it psychedelic math).

Anyways the bottom line is that if elementary particles are pure "mathematical entities", than the REAL ELEMENTARY PARTICLES ARE NUMBERS. So a virtual reality designed on a computer would be EVEN MORE REAL THAN OUR PHYSICAL REALITY because it is based on the manipulation of numbers which are even MORE FUNDAMENTAL THAN ELEMENTARY PARTICLES.

8. Nov 29, 2005

### nameta9

I'll take that back. Even if you have 2 perfect equivalent crystals, they would be slightly different because their modes of oscillation would not coincide exactly. They would have to be at absolute zero temperature, and even 2 electrons are not mathematically equal because their virtual particle clouds are slightly different. Maybe the real physical laws should be stated as two particles cannot be mathematically equivalent because quantum uncertainty makes them always slightly different.

Last edited: Nov 29, 2005
9. Nov 29, 2005

### oldtobor

MATTER - MATHEMATICS = QUANTUM UNCERTAINTY

So quantum fluctuation and uncertainty is what prevents matter/physics/universe from being a completely mathematical entity. But this uncertainty is defined mathematically. Planck's constant is a number and so is velocity of light... maybe there is a mathematical relationship between these 2 constants.

10. Nov 29, 2005

### HallsofIvy

Staff Emeritus
This has gone on for some time without anyone asking what is meant by "mathematically equal"! I wonder if someone could explain that to me.

Feynman, years ago, gave an interesting interpretation of why all electrons are "physically" identical:

Electrons can be destroyed by contact with a positron and that is the only way an electron can be destroyed. Conversely, with high enough power, an electron-positron pair can be created. An electron can never be created with a corresponding positron. A positron is an anti-partical and, in a certain sense, appears to have "negative" energy. Feynman drew a chart with "time" as the vertical axis and a single space dimension as the horizontal axis and then drew a broken line running across that.
If you cover that with a "shield" having a thin horizontal window, representing our moment in time, and move it upward you get a picture of several "dot" moving left or write in the window. We can interpret that as an electron and positron moving toward or away from each other. As we reach a downward pointing corner in the broken line the two dots come together and disappear- an electron and positron destroying each other. As we reach an upward pointing corner, we see an electron-positron pair being created.
Now remove the shield and look at the broken line. If we think the electron initially moving from left to right, moving upward (forward in time), the "pair destruction", we see a corner and the line, representing the positron moving from right to left, now moves, still to the right but downward- back in time. That is a positron is actually an electron moving backward in time!

That's why all electrons are identical- there's really only one electron in the entire universe, bouncing back and forth in time!

11. Nov 29, 2005

Staff Emeritus
It seems to me that even without having final definitons of the terms, there are good philosophical/rational reasons for wanting to distinguish "equal" from "indistinguishable". Two state vectors, rays in a Hilbert space, might be the same in the abstract, but in an interaction, each electron would be represented by its own Hilbert space and the interaction states would be in the tensor product of these two. Indistinguishable means that it doesn't matter which Hilbert space you assign to which particle, but this is conceptually a long way from equality as in 2 + 2 = 4.

Last edited: Nov 29, 2005
12. Dec 2, 2005

### net_nubie

Well, I may be missing something, but I don't see the reason for any confusion here. If you define electrons by quantum numbers, then you cannot have 2 electrons with the same quantum number in an atom. As far as free electrons go, you cannot determine there position according to Heizenberg's uncertainty principle. So you cannot say two electrons are 'mathematically' equal, because they, plain simply, are not.
Someone please tell me if i am going wrong somewhere with this.