[John369 (OP)]

This will be hard but please help me understand this pdf webhome.phy.duke.edu/~muller/Talks/Columbia_100412.pdf by answering these questions. I only need rephrasing of those technical statements in layman's terms.

1. On the 3rd page, what does it mean to implose a simplectic gauge constraints on states in D+2 dimensions? It says it renders time effectively 1-dimensional. How is it? Could 2 time dimensions always be perceived as 1D time?
2. Please explain the implications of spacetime interval on the 4th and 5th pages. How did that math work? Did we always travel in timelike trajectory and met ourselves in the past ready to start that journey? How is the worldline partly space like?
3. On the 10th page, it says motion in second time dimension is assumed to be regular, defined by initial condition, then it says small changes in initial conditions could lead to exponentially growing deviations. What is the meaning of this? How does finding a closed timelike loop become unlikely by this? Is the initial condition talking about the bold assumption that we have to move in both of the 2 dimensions all the time, or we could choose to move in one and not move in another? What is microcanonically ergodic and why should we consider one of the dimensions to be thermalized, what does it mean?
4. On the 11th page, it says energy is a two component vector in 2 timelike dimensions. How is it two component vector and what are its implications? Is it related to 5-velocity and 5-momentum and what are its implications? What are the implications of temperature becoming 2 component vector? How did temperature become 2 component vector? Please explain the whole 11th page about hot and cold time.
5. Please explain the pages from 13 to 22 in simple terms, I am specifically interested in understanding thermal stuff. How is it relating to entropy and thermal equilibrium?
6. Please explain the first two points on 25th page.
7. I can't even understand the questions posed on 29th page. Please rewrite those temperature questions in simple terms.

Please also tell me your insights on two time dimensions, and two time physics by Itzhak Bars and how he avoids stuff like closed time like loops.

[Halc]

This will be hard but please help me understand this pdf webhome.phy.duke.edu/~muller/Talks/Columbia_100412.pdf by answering these questions. I only need rephrasing of those technical statements in layman's terms.

1. On the 3rd page, what does it mean to implose a simplectic gauge constraints on states in D+2 dimensions? It says it renders time effectively 1-dimensional. How is it? Could 2 time dimensions always be perceived as 1D time?

Sorry, don't know the notation or what they're trying to convey on that page. I do notice that eliminating 2D time to 1D also seems to remove one of the spatial dimensions.

2. Please explain the implications of spacetime interval on the 4th and 5th pages. How did that math work? Did we always travel in timelike trajectory and met ourselves in the past ready to start that journey?

With multi-D time, one can create a worldline that is everywhere timelike, as illustrated with the red circle that varies only in time but not space at all.  Not sure how that's legal since the circle does move in both directions in time, and if that's allowed, why can't I define a worldline in our 3+1 space that forms a tall harrow rhombus that similarly forms a circuit without ever requiring a space-like segment of the worldline?

How is the worldline partly space like?

The blue worldline on page 4 have events separated in a space-like manner, where ds² is negative.  That never occurs in the red circle where ds² is everywhere positive.

3. On the 10th page, it says motion in second time dimension is assumed to be regular, defined by initial condition, then it says small changes in initial conditions could lead to exponentially growing deviations. What is the meaning of this?

This is standard chaos theory. In any unstable system, small differences in initial conditions (the proverbial butterfly wing flap or not) result in significant differences in eventual state.  Essentially, imagine a pencil balanced on its tip. A small difference in the initial near-vertical angle it starts out with will result in it falling in a very macroscopic different direction.

How does finding a closed timelike loop become unlikely by this?

Causality as we know it will not apply in such a situation. I don't necessarily see how this makes it less likely. It just means things don't behave in a nice causal ordered manner like all our experience shows.

Is the initial condition talking about the bold assumption that we have to move in both of the 2 dimensions all the time, or we could choose to move in one and not move in another?

I don't think there would be a 'we' that could do something that can be qualified as 'choosing' in such physics. That assumption seems to mix two different models.

As for the ergodic thermal thing, it seems that the behavior might be less chaotic if one of the time dimensions was not something anything could cleanly move through, but rather just average thermal (Brownian??) motion that averaged out to nothing at larger scales.

4. On the 11th page, it says energy is a two component vector in 2 timelike dimensions. How is it two component vector and what are its implications?

No idea. It would be nice to compare the same equations for our space so I know what sort of energy is being referenced. Energy might not be something conserved in this altered physics.

5. Please explain the pages from 13 to 22 in simple terms, I am specifically interested in understanding thermal stuff. How is it relating to entropy and thermal equilibrium?

A little above my pay grade, thank you.

The slides seem to be an accompaniment to a lecture, and do not seem to contain anywhere near the necessary information to be used as a stand-alone document. Perhaps you should find a you-tube of a presentation using this slide deck.