What is time? (Again)

[studiot]

@joigus  ;  @Markus Hanke

I suggest there is a danger inherent in considering physical time as just one mathematical coordinate axis of several say x₀, x₁, x₂, x₃  in that physical time possesses at least one property not possessed by using the mathematical notion that time is a parameter that the other axes can be put in terms of.

Time as a running parameter is excellent in many ways since it has a natural ordering that leads to causality (ie immutable sequence of events)

But if you want invariants, independent of coordinate systems as in relativity, consider what happens when you change from instance from a cartesian to a spherical coordinate system.

 

 

 

 

[joigus]

1 hour ago, joigus said:

What's the next thing you do before you engage in any calculation at all? Well, you re-define your exact solution to be "better represented" by the normal ordering:

\[:a\left(a^{+}\right)^{2}aa^{+}a^{3}a^{+}a:\overset{{\scriptstyle \textrm{def}}}{=}\left(a^{+}\right)^{4}a^{6}\]

This connection is fleshed out by a theorem (I forget the name now) that relates time-ordering of operators with expected values in the vacuum for normal-ordered products.

Irrespective of technicalities, this strongly suggests that time in getting in the way because of its being very very deeply entrenched in our language, or sequential alphabet, or what have you.

I hope it's clear what I mean, although I recognize that what I mean is difficult to make explicit in terms of language and symbols.

[joigus]

3 hours ago, studiot said:

But if you want invariants, independent of coordinate systems as in relativity, consider what happens when you change from instance from a cartesian to a spherical coordinate system.

Why do you mention spherical coordinates in relation to time? Have you got something in mind? I've got something in mind, but you go first. 

[Markus Hanke]

On 7/24/2020 at 12:04 PM, joigus said:

Another example to add to the ones you provide in which this technique wouldn't quite work are chaotic systems, because for chaotic systems there aren't nearly enough integrals of motion to reduce the dynamics to a clear trajectory that can be pictured as an implicit relation between your dynamical variables of which a one-parameter could be deduced.

Good point.

On 7/24/2020 at 12:04 PM, joigus said:

To me, once you have defined (in clear-enough cases) this one parameter that suggests to you a sequencing of events

But does it, really? I am not really sure I understand how you get from \(ds^2\) to a notion of event sequencing.

On 7/24/2020 at 12:04 PM, joigus said:

you could always define an affine parametrization for a photon which is what people do to describe the geodesic equation for photons, but the interpretation of such parameter as a time is another matter

Yes, this is essentially what I was attempting to get at - interpreting \(ds^2\) as time isn’t straightforward.

On 7/24/2020 at 12:04 PM, joigus said:

If nothing else, to me, that very strongly suggests that there's something deeply problematic about time.

I would agree to this, but like yourself I can’t at the moment put my finger onto just why that is. What’s more, I think our concept of “space” is actually equally problematic, albeit in more subtle ways. My - entirely unscientific - intuition is that neither one is really fundamental to the world, which is why I was speculating about other options over on the other thread.

On 7/24/2020 at 12:04 PM, joigus said:

I'm not saying that every mathematical statement we make has time impregnated in it, but I'm saying that it may well be that we are quite incapable of totally escaping the non-trivial consequences of having a sequential language.

Precisely - all our formal systems (languages, maths, computer code etc) are in some way sequential, because all our mental processes are. We think, feel, and reason in linear 1-dimensional ways; we can’t do anything else, because that is how our minds are structured. For me, that may (!) suggest that our models of reality (both our mental representation of the world, and our abstract descriptions of it) simply inherit that structure. In other words, this may (!) suggest that we use ‘space’ and ‘time’ in our models not because they are actual features of the world, but because that’s how our mind (the originator of those models) is structured.

So the question is, can we separate the physically relevant structures within our models (i.e. the parts that encapsulate the actual physics) from their spatiotemporal embedding? I have a feeling that we might be able to, which would have pivotal consequences for the ontological status of space and time. For example in QFT, what is the actual fundamental ontology of that framework? I don’t think it’s momentum eigenstates, or any of the operators in themselves, or the S-matrix, or even the concept of ‘particle’ - it seems we need to rather look at the commutator algebras and symmetry structures that underlie them. Which are not spatiotemporal concepts in themselves. When we play around with embeddings in different spacetimes, we then find all kinds of interesting things, like the observer dependence of vacuum states etc. This raises serious philosophical questions about space and time.

I’m just think aloud here

On 7/24/2020 at 1:23 PM, joigus said:

theorem (I forget the name now) that relates time-ordering of operators with expected values

Wick’s theorem?

 

Edited July 26, 2020 by Markus Hanke

[joigus]

1 hour ago, Markus Hanke said:

Wick’s theorem?

Yes! It's only that there were several lemmas. Wick's theorem was the re-ordering trick; and then you sandwiched the operators with the vacuum and got the ready-to-use result, which maybe had another name, an acronym like FDW or something... But Wick is one of the central names in the bunch of results.

I'll go back to your other arguments later. You make very good points.

We have similar line of thinking about one particular central question (space-time as shaped by conscience rather than fundamental). But you see an obstacle that I don't, so I'm interested. If you see an obstacle (or maybe a leap of faith or gap) it's definitely worth considering.

Another post led me into thinking about the nature of space itself, so there's food for thought there too. I posted a comment that was very naive. I had to correct myself. Measuring space is not as straightforward as I thought when I applied SR's definition based on light rays going back and forth. What about very distant objects? You must appeal to indirect methods to guess the age of stars. So, nothing simple or straightforward about that from a practical POV...

Anyway. Later.

[joigus]

2 hours ago, joigus said:

Yes! It's only that there were several lemmas. Wick's theorem was the re-ordering trick; and then you sandwiched the operators with the vacuum and got the ready-to-use result, which maybe had another name, an acronym like FDW or something... But Wick is one of the central names in the bunch of results.

Nah, that's LSZ:

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

That comes later. It's Wick's theorem I was talking about. Thank you, @Markus Hanke. +1

[Markus Hanke]

16 hours ago, joigus said:

That comes later. It's Wick's theorem I was talking about. Thank you, @Markus Hanke. +1

I wish I knew more about QFT, I never got past some ‘first introduction’ type texts, so I only know basic concepts and rough outlines. For some reason I am finding the subject difficult - not in terms of understanding the concepts, but in terms of the mathematical formalism, which I just can’t seem to really get my head around. 
Another peculiar thing about QFT is that for whatever reason it seems to set off alarm bells somewhere within me. I do not for a moment doubt its empirical success as a model, but something just seems off about it. A lot of things in it appear very ad-hoc, very messy, like an ensemble of disjoint Lego pieces that a child has put together. I just can’t, for myself, motivate the framework from fundamental considerations (as is possible to do in GR e.g.), so it seems invented and artificial. Of course I can’t offer a proper objective argument, but my intuition is telling me that we are missing something important here...something just doesn’t sit right, though I can’t put my finger on it why that is. Even the basic concept of an operator-valued field seems somehow dubious to me, and I don’t quite know why.

Of course at the moment it is the best framework we have, and it works well, but...

[joigus]

8 hours ago, Markus Hanke said:

I wish I knew more about QFT, I never got past some ‘first introduction’ type texts, so I only know basic concepts and rough outlines. For some reason I am finding the subject difficult - not in terms of understanding the concepts, but in terms of the mathematical formalism, which I just can’t seem to really get my head around. 
Another peculiar thing about QFT is that for whatever reason it seems to set off alarm bells somewhere within me. I do not for a moment doubt its empirical success as a model, but something just seems off about it. A lot of things in it appear very ad-hoc, very messy, like an ensemble of disjoint Lego pieces that a child has put together. I just can’t, for myself, motivate the framework from fundamental considerations (as is possible to do in GR e.g.), so it seems invented and artificial. Of course I can’t offer a proper objective argument, but my intuition is telling me that we are missing something important here...something just doesn’t sit right, though I can’t put my finger on it why that is. Even the basic concept of an operator-valued field seems somehow dubious to me, and I don’t quite know why.

Of course at the moment it is the best framework we have, and it works well, but...

Believe me, I share much of what you say here. And you've explained it very eloquently. +1. I'm struggling with many of these issues right now. It doesn't help that the subject is vast, also.

I have this intuitive feeling that the situation will be enormously clarified once we understand better the role that the scalar field is playing in the whole business. Right now, the scalar field is not fundamentally understood in QFT. It's used as a device to parametrize certain constraints. The scalar field comes up in just about any model-building way that people have invented to implement known facts that we don't quite understand.

Edited July 26, 2020 by joigus
minor correction

[StringJunky]

8 hours ago, Markus Hanke said:

I wish I knew more about QFT, I never got past some ‘first introduction’ type texts, so I only know basic concepts and rough outlines. For some reason I am finding the subject difficult - not in terms of understanding the concepts, but in terms of the mathematical formalism, which I just can’t seem to really get my head around. 
Another peculiar thing about QFT is that for whatever reason it seems to set off alarm bells somewhere within me. I do not for a moment doubt its empirical success as a model, but something just seems off about it. A lot of things in it appear very ad-hoc, very messy, like an ensemble of disjoint Lego pieces that a child has put together. I just can’t, for myself, motivate the framework from fundamental considerations (as is possible to do in GR e.g.), so it seems invented and artificial. Of course I can’t offer a proper objective argument, but my intuition is telling me that we are missing something important here...something just doesn’t sit right, though I can’t put my finger on it why that is. Even the basic concept of an operator-valued field seems somehow dubious to me, and I don’t quite know why.

Of course at the moment it is the best framework we have, and it works well, but...

Even if you don't progress much further than now, you've done very well for being self-taught and your ability to relate what you know is excellent. 

Edited July 26, 2020 by StringJunky

[Eise]

On 7/22/2020 at 9:58 AM, Markus Hanke said:

Mathematically, instead of having quantities that change with respect to some time coordinate, you can always have quantities that change with respect to one another, without reference to any notion of time. ‘Change’ doesn’t imply time, and time doesn’t imply change. Derivatives (in the calculus sense) with respect to some quantity other than time are well defined and commonly used.

I tend to disagree. In a graph without time, you would say that to every x belongs a value f(x). So e.g. for x = 2, f(x) = 4, for x = 3 it is 9 etc. That is a 'static' view. You can also ask for the derivative, e.g. for x= 2 f'(x) = 4, for x = 3, f'(x) = 6 etc. But that is again a static view. Only in a 'dynamic view', where you continuously change x and see how f(x) changes dependent on x, your point becomes valid. But with that, we have introduced time, namely on the x-axis.

In my opinion you are using metaphorical speech. Imagining how we 'run' through the values of x, we see how f(x) varies.

[Markus Hanke]

53 minutes ago, Eise said:

Only in a 'dynamic view', where you continuously change x and see how f(x) changes dependent on x, your point becomes valid.

This isn't how the derivative is defined, though - a quantity such as

\[\frac{df(x)}{dx}\]

does not involve any dynamics, it involves only the calculation of a static limit at a point - it is purely local. In my opinion there are no dynamics of any kind involved here, and no reference to any notion of time is implied.

Perhaps a mathematician's input would be helpful on this point @studiot

Edited July 27, 2020 by Markus Hanke

[studiot]

1 hour ago, Eise said:

I tend to disagree. In a graph without time, you would say that to every x belongs a value f(x). So e.g. for x = 2, f(x) = 4, for x = 3 it is 9 etc. That is a 'static' view. You can also ask for the derivative, e.g. for x= 2 f'(x) = 4, for x = 3, f'(x) = 6 etc. But that is again a static view. Only in a 'dynamic view', where you continuously change x and see how f(x) changes dependent on x, your point becomes valid. But with that, we have introduced time, namely on the x-axis.

In my opinion you are using metaphorical speech. Imagining how we 'run' through the values of x, we see how f(x) varies.

 

You have missed the crux of my discussion with Marcus and Michel about the teacup.

On 7/24/2020 at 9:20 AM, studiot said:

On 7/24/2020 at 8:38 AM, michel123456 said:

Another example: you walk in the desert an then enter the savanna, then the jungle. You can eventually say that the land changes (like the cup curvature) but what you are describing is not a "change in the environment", what you are describing is yourself traveling.

Your example suggests to me that you have not picked up my point.

Sorry that point is very important but also very difficult to describe/define.

Since I have not succeeded in providing enough clarity I will try again.

 

The desert, savannah etc could have separate existance, each in their own right.

As you move, they could also be changed (wink into and out of existence) like a Hollywood set.

So they are different from a cup which is only  a cup when all the parts are present together.

A handle is not a cup, a base is not a cup, a bowl is not a cup etc.

 

On 7/24/2020 at 9:20 AM, studiot said:

Yes indeed, that is why I aksed for your definition of 'change'.

Although that merely substitutes defining 'difference' for defining change.

 

Your example suggests to me that you have not picked up my point.

Sorry that point is very important but also very difficult to describe/define.

Since I have not succeeded in providing enough clarity I will try again.

 

The desert, savannah etc could have separate existance, each in their own right.

As you move, they could also be changed (wink into and out of existence) like a Hollywood set.

So they are different from a cup which is only  a cup when all the parts are present together.

A handle is not a cup, a base is not a cup, a bowl is not a cup etc.

 

But thank you for the continued discussion.

 

 

 

 

Edited July 27, 2020 by studiot

[Eise]

34 minutes ago, Markus Hanke said:

This isn't how the derivative is defined, though - a quantity such as

 

df(x)dx

does not involve any dynamics, it involves only the calculation of a static limit at a point - it is purely local. In my opinion there are no dynamics of any kind involved here, and no reference to any notion of time is implied.

Somehow I think you exactly make my point. No dynamics, therefore no change. 

[joigus]

1 hour ago, Eise said:

I tend to disagree. In a graph without time, you would say that to every x belongs a value f(x). So e.g. for x = 2, f(x) = 4, for x = 3 it is 9 etc. That is a 'static' view. You can also ask for the derivative, e.g. for x= 2 f'(x) = 4, for x = 3, f'(x) = 6 etc. But that is again a static view. Only in a 'dynamic view', where you continuously change x and see how f(x) changes dependent on x, your point becomes valid. But with that, we have introduced time, namely on the x-axis.

In my opinion you are using metaphorical speech. Imagining how we 'run' through the values of x, we see how f(x) varies.

The technique that Markus describes is actually something I've seen done in theories of gravity when people try to discuss emergent time. But I think you're right that the sequencing aspect is lost or not entirely obvious. That's why I suggested the introduction of a proper length parameter as the most likely candidate to where, mathematically, time comes from.

Suppose you have a physical system with variables (x₁,x₂,...,x_N). If you had N-1 implicit equations or "constraints":

f₁(x₁,...x_N)=0

f₂(x₁,...x_N)=0

...

f_N-1(x₁,...,x_N)=0

This would amount to having a common history for all the x's. The complete deterministic solution to a particular system's evolution deployed before your eyes. Giving values to all of them except one allows you to infer the value of the remaining one. Although that value may not be unique. For example. At a certain point in your life you asked your sweetheart to marry you. That's x₇=1. But there are two values for the answer that she gave you: x₈=1 and x₈=0. That's because at some point she said "no", you asked again a week later, and she said "yes". Assigning values to the rest of the variables of the universe, you can get "back" to the event you want to picture. This gives you a bird's eye view of the history.

If you have a metric for the x's, you can do something quite impressive in principle if your metric is positive definite and based on a quadratic form. You build the infinitesimal interval along your implicit curve:

ds²=g_ijdxⁱdx^j

and define your sequencing (primitive time) as either,

dt=+sqr(g_ijdxⁱdx^j)

or

dt=-sqr(g_ijdxⁱdx^j)

I know @Markus Hanke has some exceptions/criticism for this procedure. But I think it's a mathematical language that very much fits what we know about the mathematics of the world, has many of the essential features of time in it, and leads to a quite clear-cut setting (if not solution) of the problem: Why is it that we only see the world evolve in one of the two orientations that the metric allows you to pick? The mathematics seem to confirm that: You chose one or the other, but once you're fixed on one, no valid re-parametrization can give you the other.

I have an intuition that this may have to do with the Michel/Studiot argument about the cup of tea. But I must go over their arguments again, I must say.

Edit: x-posted with Eise

Edit 2: The metric should be defined on the f's, not the x's. So it'd be,

ds²=g_ijdfⁱdf^j

Edited July 27, 2020 by joigus
minor addition/correcting mistake

[Eise]

34 minutes ago, studiot said:

You have missed the crux of my discussion with Marcus and Michel about the teacup.

I am afraid I really do not see the crux. 

[Markus Hanke]

19 minutes ago, Eise said:

Somehow I think you exactly make my point. No dynamics, therefore no change.

Well, for me this quantity describes the local relationship between neighbouring points. I think it is really a matter of definition whether this qualifies as 'change' or not. For me it does, without necessitating any reference to time.

[studiot]

1 hour ago, Eise said:

I am afraid I really do not see the crux. 

Thank you for the response.

It is difficult to explain further without my knowing what it is you don't see.

Normally you offer a well reasoned criticism to discuss.

 

Edit Marcus' still only refers to space (and time).

But there is no reason for what I call the running variable (or either variable) in a graph to include time.

Good examples of other running variables are temperature and composition.

A eutectic diagram is a combination of both, neither space nor time variables get a look in.

A simpler phase diagram might even make Marcus point more forcefully since 'neighbouring points' could be in different states  - say solid and liquid.

 

As I like to keep pointing out,

Nature is more artful than the best straightjacket human rules can determine.

Edited July 27, 2020 by studiot

[Markus Hanke]

54 minutes ago, studiot said:

Edit Marcus' still only refers to space (and time)

I think in a very general sense, I am referring to the relationship between quantities, regardless of what they are - if there is a relationship other than identity between them, I see no reason why one quantity can't change with respect to another one without needing any temporal dynamics. For spacetime in particular, 'space' and 'time' are on equal footing anyway, so if something can 'change' with respect to time, it can also 'change' with respect to any other quantity. But like I said, I suppose in the end it is a matter of definition / convention.

[michel123456]

22 hours ago, studiot said:

 

You have missed the crux of my discussion with Marcus and Michel about the teacup.

 

 

 

 

 

I understand what you mean.

Yes, your tea cup has different curves, and it happened that I didn't know why I should argue against your argument. It makes sense. The curve changes in relation with the spatial coordinates. No time involved.That seems OK

But after more thinking when you are stating "when all the parts are present together", I suspect that you mean "all the parts are present together at the same time". Or maybe (I don't know to say otherwise) that your cup somehow is existing there in space "out of time". As if the concept of a cup "existing" could be understood without the concept o time. But the concept of "together" has already the idea of time inside it. When the bus hits you, it is because you shared the same space together at the same time.

Because together at different times makes no sense. If you share the same space at different times, you are not "together".

And after more profound thinking, in your statement, there is time 4 times: when all the parts are present together

_when

-are

_present

-together

20 hours ago, studiot said:

Good examples of other running variables are temperature and composition.

Good point.

Maybe inside the concept "variable" there is also time hidden.

Edited July 28, 2020 by michel123456

[Markus Hanke]

2 hours ago, michel123456 said:

But after more thinking when you are stating "when all the parts are present together", I suspect that you mean "all the parts are present together at the same time". Or maybe (I don't know to say otherwise) that your cup somehow is existing there in space "out of time".

My original thought experiment was about a tea cup in a hypothetical (!) universe that only had 3 spatial dimensions and no temporal ones (and isn’t embedded into anything higher-dimensional), so speaking of “simultaneity” is meaningless there. You can only slice up such a universe into slices that are purely spatial. Studiot’s choice of words ‘present together’ was very apt in that regard - two points are ‘present together’ if they share the same spatial slice. Crucially, on a manifold that is purely 3D, all pairs of arbitrary points share a common spatial slice, so the entire manifold is ‘present together’. At the same time, two arbitrary points A and B are generally not identical, because geometrically you have \(ds^2 \neq 0 \) unless A=B.

So essentially, my way of thinking is this - if you have a pair of entities of the same type {A,B} which are not related by an identity relation, then this defines a notion of ‘change’, to be understood as a relationship between A and B. ‘Entity’ (I couldn’t think of a better term) is to be taken in its most general sense. To put it very succinctly - change is simply the failure of entities of the same type to be identical. Not only does defining it like this avoids any reference to time, it also does not make reference to space; change (again, in my thinking) is not fundamentally a spatiotemporal concept at all. There is no reason why it needs to be, other than our human intuition, which is not a very objective criterium. I think it is better understood as a set-theoretical notion. Yes, you can apply it to points on a differentiable manifold, but you can also apply it to any other kind of entity, be it mathematical or not.

For the specific example of the tea cup, evaluating some measure of curvature (e.g. Gaussian curvature) at the handle and at the base of the cup will yield different results in general, so the relationship between them is one of non-identity - so there is a notion of change of Gaussian curvature with respect to spatial coordinates, even though the manifold is purely 3D.

I make it clear again that this is only my own thinking on this matter - personal philosophising, if you so will. It is not intended as any kind of claim - it’s more of a ‘thinking aloud’ kind of thing.

 

 

Edited July 28, 2020 by Markus Hanke

[Eise]

22 hours ago, studiot said:

It is difficult to explain further without my knowing what it is you don't see.

It is also difficult to explain what I don't see, if I don't see it... 

However, I see something else:

22 hours ago, studiot said:

But there is no reason for what I call the running variable (or either variable) in a graph to include time.

Good examples of other running variables are temperature and composition.

Time is implicit in 'running'. I keep thinking about those math video's where they animate this 'running': A point is moving along the x-axis, and on the y-axis a point moves according to f(x). And in my opinion, that is the point where also the word 'change' can be used. We change the values of x, and show how the value of y changes. So time is implicit when we say 'y changes according to the changes of x'.

[studiot]

2 hours ago, michel123456 said:

I understand what you mean.

But after more thinking when you are stating "when all the parts are present together", I suspect that you mean "all the parts are present together at the same time". Or maybe (I don't know to say otherwise) that your cup somehow is existing there in space "out of time". As if the concept of a cup "existing" could be understood without the concept o time. But the concept of "together" has already the idea of time inside it. When the bus hits you, it is because you shared the same space together at the same time.

Because together at different times makes no sense. If you share the same space at different times, you are not "together".

And after more profound thinking, in your statement, there is time 4 times: when all the parts are present together

 

I'm glad you got the first point, and what's more you agree with it.

I also understood your very good point about the savannah / desert example.

But this is a different point so I am sorry you missed my second point.

Whilst it is possible that savannah and desert coexist side by side so you could make the journey from one to the other as you describe.
Time would then necessarily be involved o that when you were in the desert, the savannah still existed and vice versa.

It is also possible to that only one landscape exists and for a transformation from savannah to desert (or vice versa) as you pass from one area to another.

In both cases a change is involved, but this is why I said a transformation is not the only form of change.

 

However in the case of the cup, it is not possible for the bowl to transform into a handle as you look from bowl to handle.
No cup can exist under such circumstances.
Both handle and bowl must be present for the cup to have existance.

Yet there remains a change of curvature.

 

I also offerd a more complicated example to which you replied

2 hours ago, michel123456 said:

Good point.

Maybe inside the concept "variable" there is also time hidden.

But you did not elaborate.

If I were to further offer you some (hidden) H₂O at 0^o C it would be impossible for you to tell if that water was solid or liquid, or even a mixture of both, no matter how much time you has available.

The answer to the question simply does not depend upon time in any way.

 

As Marcus and I said, It must depend upon your definition.

If you are going to adopt a definition based upon change then I consider I have shown examples of change that do not depend another variable than the one stated.

The curvature of the object depends upon position only.

The possible states of matter depend upin the temperature only for the eutictic and the cooling substance.

Edited July 28, 2020 by studiot

[Markus Hanke]

6 minutes ago, Eise said:

We change the values of x, and show how the value of y changes.

But what if we think of f(x) as an (uncountably) infinite set of real numbers (which it is, mathematically speaking)? Or better still - a 1-parameter family of real numbers? That set would be a static construct, as would be the relationship between the elements in the set. Of course you could externally impose a notion of “going from one element to another”, but I don’t think that is inherent in the set itself.

[studiot]

10 minutes ago, Eise said:

Time is implicit in 'running'. I keep thinking about those math video's where they animate this 'running': A point is moving along the x-axis, and on the y-axis a point moves according to f(x). And in my opinion, that is the point where also the word 'change' can be used. We change the values of x, and show how the value of y changes. So time is implicit when we say 'y changes according to the changes of x'.

You are describing what is mathematically called a 'locus'  - The path traced out by a point moving under certain conditions.

That is different from a plot of a graph.

[Markus Hanke]

Just now, studiot said:

You are describing what is mathematically called a 'locus'  - The path traced out by a point moving under certain conditions.

Interesting, I haven’t heard this term before!