# Yang–Mills and Mass Gap

Suppose you construct a set of parallel lines in $$\mathbb {R}^2$$. Now you mark regularly spaced points along the lines and map the points in adjacent lines to each other. Does it matter if different lines have different spacing? (yes it does, in $$\mathbb {R}^2$$ because integer arithmetic is based on the integers being separated by unit lengths).

Ok, so suppose you start with a square lattice of points in $$\mathbb {R}^2$$; How many ways can you construct sets of parallel lines, so you can map points to points between adjacent parallel lines?

I've already given at least one example earlier, but sure, here's another one: let's say a sphere rotates once per hour. 1 hours is 60 minutes, so $$T=60$$, which is larger than $$1$$. QED.

Is $$60 minutes > 1 hour ?$$

You indeed seem to think so.
So, maximum $$T$$ is $$1$$ unit of time.

So, maximum $$T$$ is $$1$$ unit of time.
That is indeed your claim. But your claim is inconsistent: if $$T$$ can at maximum be $$1$$ hour, then it's maximum is also $$60$$ minutes, and thus larger than $$1$$. In other words, if $$T$$ has a maximum of $$1$$, it can have a value of $$60$$, which is a logical contradiction, and thus your claim is proven false. (Tip: Look up "proof by contradiction".)

That is indeed your claim. But your claim is inconsistent: if $$T$$ can at maximum be $$1$$ hour, then it's maximum is also $$60$$ minutes, and thus larger than $$1$$. In other words, if $$T$$ has a maximum of $$1$$, it can have a value of $$60$$, which is a logical contradiction, and thus your claim is proven false. (Tip: Look up "proof by contradiction".)
Hansda, have you finally realized the error you have made?

Hansda, have you finally realized the error you have made?

Consider any time duration/interval as $$t$$units of time. Consider time-period of oscillation as $$T$$units of time. Here to get atleast one complete oscillation within the time interval $$t$$ units, $$T\leq t$$ . Here frequency $$f=\frac{t}{T}$$ or $$fT=t$$ .

And now substitute $$t$$ with $$2t$$. Note that this means that $$T$$ has to be substituted with $$\frac{T}{2}$$ to keep the physics the same. So if $$T=1$$ is the minimum, so is $$T=2$$, etc. Your claim of a minimum $$T$$ of $$1$$ is thus incoherent.

And now substitute $$t$$ with $$2t$$.

Here $$t$$ is any unit of time. Why do you want to substitute $$t$$ with $$2t$$. Seems you are trying to twist my statement and make it complicated.

Note that this means that $$T$$ has to be substituted with $$\frac{T}{2}$$ to keep the physics the same.

Just your imagination. There is no need of it.

So if $$T=1$$ is the minimum, so is $$T=2$$, etc. Your claim of a minimum $$T$$ of $$1$$ is thus incoherent.

Again your imagination. I am not claiming $$T=1$$ is the minimum.

Here $$t$$ is any unit of time. Why do you want to substitute $$t$$ with $$2t$$.
Because for that formula to mean anything physical, it must be true independent of the unit of choice. That's basic physics.

Seems you are trying to twist my statement and make it complicated.
No, I'm trying to make you understand why you are wrong.

Just your imagination. There is no need of it.
There is only no need, if you think there's no need to make sense.

Again your imagination. I am not claiming $$T=1$$ is the minimum.
Sorry, I keep mixing it up. You think $$T=1$$ is a maximum. Just throw in some inverses in the right places.

Because for that formula to mean anything physical, it must be true independent of the unit of choice. That's basic physics.

Here $$t$$ is having, any unit of time. That means it is independent of unit of choice.

Sorry, I keep mixing it up. You think $$T=1$$ is a maximum. Just throw in some inverses in the right places.

I am also not claiming this. This is your another imagination. You can re-read my post #148.

Here $$t$$ is having, any unit of time. That means it is independent of unit of choice.
But its numerical value isn't; that's my point. Look: 1 hour = 60 minutes. By changing the unit of time, I have to change the numerical value too. So when you say $$f\geq 1$$ without specifying the units, it's meaningless. Do you mean: $$f\geq 1 s^{-1}$$? Or $$f\geq 1 hour^{-1}$$? Your original claim is nonsensical.

I am also not claiming this. This is your another imagination. You can re-read my post #148.
Post #54: You claim that the minimum of frequency is $$1$$.
Post #97: You say that $$fT=1$$, and thus that $$f\geq\frac{1}{T}$$
Combine the two:
$$f=\frac{1}{T}\geq 1$$
Invert it:
$$\frac{1}{f}=T\leq 1$$

QED

But its numerical value isn't; that's my point. Look: 1 hour = 60 minutes. By changing the unit of time, I have to change the numerical value too. So when you say $$f\geq 1$$ without specifying the units, it's meaningless. Do you mean: $$f\geq 1 s^{-1}$$? Or $$f\geq 1 hour^{-1}$$? Your original claim is nonsensical.

Post #54: You claim that the minimum of frequency is $$1$$.
Post #97: You say that $$fT=1$$, and thus that $$f\geq\frac{1}{T}$$
Combine the two:
$$f=\frac{1}{T}\geq 1$$
Invert it:
$$\frac{1}{f}=T\leq 1$$

QED

Ah, I see you fail to understand the quoting system:
- Look very carefully at those pieces of texts that are displayed in boxes: they were originally written by you! You can even tell they came from post #152. So that's the post I'm replying to.
- I see I didn't quote your post #148 in my reply #149. Typically, if no post is quoted it's either a stand-alone post (which clearly isn't the case here), or a reply to the post directly above. So my reply to your post #148 is post #149.

Here time duration $$t$$ can have any unit of time. $$t$$ can be any amount of time. So substituting $$t$$ with $$2t$$ is unnecessary.

Here time duration $$t$$ can have any unit of time. $$t$$ can be any amount of time. So substituting $$t$$ with $$2t$$ is unnecessary.
No, you are misunderstanding. I'm not changing the physical time, I'm changing both the numerical value and the unit. 1 hour = 60 minutes. Also, something being unnecessary doesn't mean it can't be done.

No, you are misunderstanding. I'm not changing the physical time, I'm changing both the numerical value and the unit. 1 hour = 60 minutes. Also, something being unnecessary doesn't mean it can't be done.

This way, what you are going to achieve? What you think about my post #148. It is right or wrong?

This way, what you are going to achieve?
I'm trying to make you understand that the statement $$f\geq 1$$ you made is incoherent.

What you think about my post #148. It is right or wrong?
I'm trying to make you understand that the statement $$f\geq 1$$ you made is incoherent.