Understanding Cosmology (Post 1)

[Polymath257]

@LegionOnomaMoi and @Polymath257 over here nerding out on tensors, while us physicists over here are like "hur hur we can make bra jokes with <bra|kets>"

(I'm enjoying the discussion)

Don't get me started.....

Those are just the distinction between a vector space and its dual. In the case of Hilbert spaces, you can identify the dual and the original space.

Why use <x|y>? In a Hilbert space, it is just <x,y>. But x and y are both vectors. There is no reason to consider <x| and |y>. The only distinction is whether you are thinking of the vector as in the original space or an element of the dual. But since they are isomorphic, it doesn't matter. Just use x and y.

In other words, in most cases, bras and kets are unnecessary.

 

[Meow Mix]

Don't get me started.....

Those are just the distinction between a vector space and its dual. In the case of Hilbert spaces, you can identify the dual and the original space.

Why use <x|y>? In a Hilbert space, it is just <x,y>. But x and y are both vectors. There is no reason to consider <x| and |y>. The only distinction is whether you are thinking of the vector as in the original space or an element of the dual. But since they are isomorphic, it doesn't matter. Just use x and y.

In other words, in most cases, bras and kets are unnecessary.

This is why we need mathematician friends. We have been taught basically “does it have a vector and a linear operator, and are you doing something remotely quantum-y? Better slap some bra-kets in there.”

 

[Polymath257]

This is why we need mathematician friends. We have been taught basically “does it have a vector and a linear operator, and are you doing something remotely quantum-y? Better slap some bra-kets in there.”

One thing that has helped me greatly is thinking of components as evil. Sometimes they are necessary for calculations, but for general discussion, they hide the simplicity of the concepts.

 

[Polymath257]

This is why we need mathematician friends. We have been taught basically “does it have a vector and a linear operator, and are you doing something remotely quantum-y? Better slap some bra-kets in there.”

One difficulty is that physicists and mathematicians use such different definitions for the same basic concept that both sides are thrown off by the way the other does things.

I started out in physics. There was one specific physics book (Gravitation by MTW) that made me decide I needed to go into math just to figure out what was going on. I went on to get a PhD in abstract harmonic analysis.

But, after I got tenure, I went back and took physics courses. I passed the PhD qualifying exams in physics and was going to get my PhD studying dark matter, but my advisor died and the backup advisor moved. At the same time, my responsibilities in the math department increased, so the PhD never happened.

 

[LegionOnomaMoi]

Don't get me started.....

Those are just the distinction between a vector space and its dual. In the case of Hilbert spaces, you can identify the dual and the original space.

Why use <x|y>? In a Hilbert space, it is just <x,y>. But x and y are both vectors. There is no reason to consider <x| and |y>. The only distinction is whether you are thinking of the vector as in the original space or an element of the dual. But since they are isomorphic, it doesn't matter. Just use x and y.

In other words, in most cases, bras and kets are unnecessary.

I am reminded of Mermin's words here, and how I side completely with the mathematicians on this:
"Mathematicians tend to despise Dirac notation, because it can prevent them from making important distinctions, but physicists love it, because they are always forgetting that such distinctions exist and the notation liberates them from having to remember" (http://people.cs.clemson.edu/~steve/CW/395/CS483-part1.pdf)

 

[LegionOnomaMoi]

The determinant map for a vector space

Determinants predate vector spaces and indeed the entirety of matrix algebra by a substantial amount of time. Nor did I specify that I meant by "determinant" a "determinant map for a vector space", because apart from anything else we still (horror or horrors) teach undergraduate mathematics students that one can think of cross products and such in terms of determinants (when, as a multilinear functional, it makes no sense to speak of them as such). Regardless, you are bordering on a trivialization here that is beyond that which I made in jest. Certainly, one can refer to the real number line in terms of many algebraic, topological, and geometrical concepts that an intuitive sense of this same number line was used in order to generalize from. It makes very little sense to take issue with referring to tensors as things that transform as tensors only to essentially fail to uniquely define tensors except by virtue of some properties which will only hold for tensors in particular cases while insisting that the distinctions between vectors, forms, determinants, etc., are somehow relatively unimportant compared to the definition I gave of tensors in jest and your "correction."

Not what I am talking about.

I'm not sure what you're talking about. I made a joke I first learned from a string theorist and have encountered subsequently since, and you came back with a correction that must, by definition, either be wrong or not a correction at all. I said

That's easy. A tensor is something that transforms like a tensor.

You replied:

No, actually, a tensor is just a multilinear map

So, as long as we're being overly formal here, you are disagreeing with me that a tensor is something that transforms like a tensor. Please supply an example of a tensor that doesn't transform like a tensor.

 

[LegionOnomaMoi]

One thing that has helped me greatly is thinking of components as evil. Sometimes they are necessary for calculations, but for general discussion, they hide the simplicity of the concepts.

Yes, a point made clear in both of the lecture series I posted, e.g., here (or see more generally but in a shorter series here: Lecture 3: Multilinear Algebra [International Winter School on Gravity and Light 2015])

 

[Polymath257]

I am reminded of Mermin's words here, and how I side completely with the mathematicians on this:
"Mathematicians tend to despise Dirac notation, because it can prevent them from making important distinctions, but physicists love it, because they are always forgetting that such distinctions exist and the notation liberates them from having to remember" (http://people.cs.clemson.edu/~steve/CW/395/CS483-part1.pdf)

I'd actually say the opposite. The Dirac notation tends to make distinctions that there is no good reason to make. What is the difference between <x| and|x>? In actuality, nothing. In both cases, you have a vector in a Hilbert space. In one, you regard it as a linear functional (via Riesz' theorem) and on the other, you don't. But which is which? What difference does it make? Ans: none at all.

And, yes, properly chosen notation should make it easier to ignore irrelevant distinctions, such as when we use a standard isomorphism to identify a Hilbert space and its dual. The notation can even be nicely extended whenever you have a type of inner product between two vector spaces.

So, as long as we're being overly formal here, you are disagreeing with me that a tensor is something that transforms like a tensor. Please supply an example of a tensor that doesn't transform like a tensor.

If your transformation equation has any partial derivatives in it, then you miss a lot. Those are just one type of basis transformation. Also, if you only have tangent vectors and their duals as entries, you also miss a lot. Many tensors do not live on manifolds and do not require those restrictions on the entries (they can come from any number of vector spaces, potentially of infinite dimension).

But I can see the audience getting tensor and tensor.

 

[Meow Mix]

One difficulty is that physicists and mathematicians use such different definitions for the same basic concept that both sides are thrown off by the way the other does things.

And often physicists (as you and Legion have been noting) simplify to the point of lacking mathematical distinctions, which leads to a lack of nomenclature that mathematicians would use.

For instance this leads to just assuming a mathematician using a word in an unfamiliar way is a new concept (“oh, one of those math people things”) rather than the same thing we physics people know and love under a different or extended name.

 

[Meow Mix]

For example, I’m totally guilty of what Legion was talking about re: thinking of cross products in terms of determinants. But most of my math has been undergraduate.

The way my school has worked is the graduate level “math for physicists” courses are probably dumbed down for physicists, they come in the form of classes called “Methods of Theoretical Physics” and such which are basically “hey, here’s some linear algebra and diffey q, but only the bits you’ll probably use.”

 

[Polymath257]

And often physicists (as you and Legion have been noting) simplify to the point of lacking mathematical distinctions, which leads to a lack of nomenclature that mathematicians would use.

For instance this leads to just assuming a mathematician using a word in an unfamiliar way is a new concept (“oh, one of those math people things”) rather than the same thing we physics people know and love under a different or extended name.

And it is true that mathematicians have different goals. We tend to explore the abstract ideas and not worry about applications. We also require a much higher standard of proof of the ideas.

Physicists, on the other hand, are more interested in the real world (as they should be) and the strict mathematical idea of proof is unnecessary simply because the ideas have a different standard: that of testability and agreeing with observations.

But what I have found is that physicists tend to use the ideas and notation that were common in the late 19th century when doing general relativity. That was overly concerned with components of vectors and tensors. In the intervening century, mathematicians have cleaned up these ideas, simplified them, and unified them in ways that, I believe, could greatly help the discussions in physics.

One example is that of the Christoffel symbols. For most mathematicians, these have been replaced by the notion of a connection. A pseudo-Riemanning metric will select a particular connection for which the metric is invariant. But, and this is relevant in QM, there are other connections than this one and those can be relevant for modeling particle fields.

And, again, it isn't how the components change under a change of basis that is the key: it is a coordinate-free description that simplifies the analysis and clarifies the ideas. The real key is the differential operator that arises. Then, after you choose a coordinate system, the components can be dealt with easily.

Think of the simplification obtained by doing Maxwell's equations in vector form as opposed to writing out the formulas for the different components. Instead of four nice equations, the component version has 8 equations that are not nearly as compact nor clear. It gets even better when spacetime tensors are used: Maxwell's equations then reduce to two tensor equations.

And, once you have that, you can choose the coordinates you want, similar to deciding to have the E&M wave be along the z-axis. it simplifies the analysis and loses no generality. But, you don't worry about how the components would change if the direction was along some other vector. You go back to the vector equations and write them down directly in the new coordinate system.

 

[Polymath257]

For example, I’m totally guilty of what Legion was talking about re: thinking of cross products in terms of determinants. But most of my math has been undergraduate.

The cross product is unusual because it only really works in three dimensions. And the reason is that it really isn't a vector but, wait for it, an alternating product of forms. THAT sort of product can be done in any number of dimensions, but the product of two 1-forms is a 2-form. Only in three dimensions is the vector space of 2-forms the same dimension as the vector space of vectors.

This also explains why you get a 'pseudo-vector' as opposed to a vector for the cross product.

The way my school has worked is the graduate level “math for physicists” courses are probably dumbed down for physicists, they come in the form of classes called “Methods of Theoretical Physics” and such which are basically “hey, here’s some linear algebra and diffey q, but only the bits you’ll probably use.”

Indeed. And the deeper reasons those special functions come up are never really mentioned.

For example, the spherical harmonics are much better described as the components of representations of the group of rotations. As such, they will arise whenever there is a rotationally symmetric situation. Any rotationally symmetric linear PDE will have solutions that can be broken down into spherical harmonics (for the angular component) because of this.

Bessel functions arise naturally as radial components for any rotationally symmetric situation because they are essentially the radial component of the Fourier transform of measures living on the circle (or sphere). The dimension in which you do the Fourier expansion determines the order of the Bessel function.

I can go on, but I hope the point is made (or at least suggested).

 

[Heyo]

For example, I’m totally guilty of what Legion was talking about re: thinking of cross products in terms of determinants. But most of my math has been undergraduate.

The way my school has worked is the graduate level “math for physicists” courses are probably dumbed down for physicists, they come in the form of classes called “Methods of Theoretical Physics” and such which are basically “hey, here’s some linear algebra and diffey q, but only the bits you’ll probably use.”

"Physics is becoming too difficult for the physicists." - David Hilbert

 

[Meow Mix]

I can go on, but I hope the point is made (or at least suggested).

Yes. Just seems like there's so much to keep track of. I feel like I'm going to need to work more on the math side rather than calling what I get from classes and projects good enough.

We did spherical harmonics in E&M 1 & 2 but it's just sort of presented as "here's how to solve this," not a lot of deep understanding of what's mathematically happening. I mean we did get a little bit of that in the Methods classes but not a lot. Plus that was a little while ago now. I still have my notes, but for instance I just remember doing stuff that I didn't understand why we were doing it at the time, like I remember using matrices and Legendre and Laguerre polynomials for... honestly reasons I don't remember at this point (I have just taken out my notes, and it looks like these were chosen examples for some linear transformations).

This just gives me anxiety, imposter syndrome is already bad enough, ha! I think I'll make it a point more often when I use a thing to really deeply get into that thing, or ask a math friend about it. It can be hard to get an understanding as a physics person because it's like you're climbing a ladder outside a building and just peeking in through windows without knowing the entire foundation of the building. So going and picking up a math text (which I've thankfully at least done a couple of times) can be hard because if you get to a chapter with the thing you want to know about, you might get stuck thinking "uh, what?" (Due to dense terminology, unfamiliar [compared to physics] syntax, etc.)

 

[LegionOnomaMoi]

I'd actually say the opposite. The Dirac notation tends to make distinctions that there is no good reason to make. What is the difference between <x| and|x>?

Quite a bit, actually. Both physically and mathematically, despite Riesz' lemma (or theorem). But as I've hijacked this thread enough, I am responding to this only because I think that the material I can link to here may be of some use and/or relevance. Also, I've found that (personally and professionally) Schuller is a great source for clarity when it comes to the structures of theoretical physics whether one is a mathematician interested in physics or a physicist interested in gaining greater mathematical insights, clarity, etc.
To that end:

I'd recommend watching the entirety of the lecture series, not just the lecture, but for conciseness the most relevant portions (after the introductory piece) begin at ~1:27:00, which (as does the introduction) addresses the importance and remaining issues with the use of the Riesz' lemma.

 

[Polymath257]

Quite a bit, actually. Both physically and mathematically, despite Riesz' lemma (or theorem). But as I've hijacked this thread enough, I am responding to this only because I think that the material I can link to here may be of some use and/or relevance. Also, I've found that (personally and professionally) Schuller is a great source for clarity when it comes to the structures of theoretical physics whether one is a mathematician interested in physics or a physicist interested in gaining greater mathematical insights, clarity, etc.
To that end:

I'd recommend watching the entirety of the lecture series, not just the lecture, but for conciseness the most relevant portions (after the introductory piece) begin at ~1:27:00, which (as does the introduction) addresses the importance and remaining issues with the use of the Riesz' lemma.

If anything, this lecture supports my point. The Dirac notation only makes things more complicated. I'll watch other lectures at another time.

I briefly looked at a couple of other lectures. They seem like a standard overview of Functional Analysis.

 

[Meow Mix]

Gonna write Post 6 and introduce dark energy this week y'all and I have to somehow describe cosmological equation of state parameters in ways laypersons would understand. I kind of feel the pop sci peoples' pain now when they try to write books.

 

[Polymath257]

So, as long as we're being overly formal here, you are disagreeing with me that a tensor is something that transforms like a tensor. Please supply an example of a tensor that doesn't transform like a tensor.

the point is that the tensor doesn't transform at all. it is the same throughout.

What transforms are the *components* of the tensor. And those change based on the change of basis vectors used in finding the components.

And, since tensors do not have to be based on a single vector space (think two particle systems), you may need a change of basis in each factor of the tensor product. In the standard case of a single vector space and its dual, a single change of basis also provides a change of basis for the dual. But that is a very special case.

Furthermore, even if dealing with vector and tensor *fields* on manifolds and when dealing with a change of basis obtained by changing charts, it is quite possible, and even useful, to have the vectors be other than simply tangent vectors. Which means that the differential form for the change of basis need not even apply.

 

[Meow Mix]

I have a lot if catch-up posting to do. I owe some people responses, will probably try to take care of that first.

Then the next post on dark energy will commence.

 

[Truthseeker]

@Meow Mix, this is under science and religion? They should have a category called "science" alone.

Edit: There is a category called science and technology, I suggest to the mods to move this there

I've read some of the posts here early on. Some I can understand, some not. @JoshuaTree seems to have some Physics understanding beyond what I have, let alone you.

I'm late to this series, I think I'll have to let it go! There are already 6 groups of twenty here, and there are seven such threads. Thanks, anyway.