Symmetry is kind of a big deal in physics — big enough that the magazine jointly published by the SLAC and Fermilab accelerator laboratories is simply called symmetry. Symmetry appears in a variety of contexts, but before we dive into them, we have to understand what “symmetry” actually means. Which is what we do in this video, where we explain the basic ideas of what mathematicians call “group theory.” By the end you’ll know exactly what is meant, for example, by “SU(3)xSU(2)xU(1).”
The Biggest Ideas in the Universe | 14. Symmetry
And here is the associated Q&A video:
The Biggest Ideas in the Universe | Q&A 14 - Symmetry
In your presentation you mentioned Noether’s theorem. Let us assume some ingenious physicist came up with a ToE today, which symmetries would we expect it to have? Or put the other way round: In an expanding universe that is driven apart by dark energy – whatever that may turn out to be – would we expect conventional conservation laws to be still valid on the largest scale?
Charge parity violation seems to be a big deal and the T2K collaboration recently reported indications of this asymmetry for leptons. What are the implications?
I’m enjoying this series of talks very much because it gives us an idea of the real nuts and bolts of physics and somehow you make it good humoured and entertaining. Thank you.
Hi Sean, thanks for the talks on symmetry!
Some questions I have of which I’m not sure I can even ask these questions:
1) The representation of a 1D Complex axis is a 2D plane or it IS a 2D plane?
2) Our space can be represented (or is?) as a 3D Real vector space, but all the underlying fields are Complex. So, quantumfields have more complex “options” per unit of real space?
3) Could you sum/integrate/add/… Non-continuous groups in some way to become continuous groups? Or is this the same as evaluating the symmetry group for each step closer to something continuous. (f.e.: D3 for triangle……. -> D5 pentagon….. -> ?? for Circle -> ??).
4) The first law of thermodynamics states that energy is conserved. But sometimes nature can indebt itself in energy currency by allowing virtual particles to generate and annihilate. Could our universe be a massive “loan” of energy that’s bound to be repaid in the future? Or is the total amount of energy in equilibrium the lowest possible amount of energy, and only this is allowed to not be repaid? In other words: Do these symmetries hold inside the universe but not for the entire universe?
Mathematicians come up with all kinds of strange things. Have they found something that’s not symmetrical under the Identity transformation?
A silly comment, perhaps, on the subject of abstract mathematics.
It seems the square root of -1 (imaginary numbers) has been enormously productive. Yet, “division by zero” is “not allowed”.
I’m curious about this. I’m guessing it hasn’t been “productive.” I’m summarizing the usual objections in that way which I know. I suppose “infinity” close to zero is one way out… but still.
Next time I’ll have a more focused response to your video talks.
Keep up the good work! Many thanks.
Why is one rotation or flip followed by another defined as multiplication rather than addition?
Can you discuss continuous scale symmetry in your Q&A? Namely:
=> https://www.newyorker.com/tech/annals-of-technology/einsteins-first-proof-pythagorean-theorem
=> https://youtu.be/s2_8grzsYvw?t=125
=> https://en.wikipedia.org/wiki/Metallic_mean
=> https://youtu.be/smCDtjDLdf8?t=130
If you can explain how physicists use scale symmetry for exponentiated integer counting — computational tile recursion, vanishing point on the horizon, generating functions, creeping up to the Infinitesimal — please do!
=> https://en.wikipedia.org/wiki/Cellular_automaton
=> https://en.wikipedia.org/wiki/Perspective_(graphical)
=> https://en.wikipedia.org/wiki/Formal_power_series
Thank you for this elucidating video series!
Thanks for the links, Kyle. All interesting!
Thanks William!
Here’s another one => https://en.wikipedia.org/wiki/Adequality
“Legal triage” amputates the end of a function’s output too long for practicality.
=> https://en.wikipedia.org/wiki/Long_tail
=> https://en.wikipedia.org/wiki/Rule_against_perpetuities
=> https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
I read your question above about zero.
I really enjoyed The Book of Nothing by John D. Barrow.
I don’t know about division by zero, but I did a lot of R&D on zero to figure out how to design a JavaScript object tree with function object nodes included => https://seedtreedb.com
From my layperson POV, zero seems to be the one integer firmly rooted in “quantum logic” properties, especially Brahmagupta’s zero or maybe even Conway’s Surreal Numbers.
We’ve learned a lot about quantum logic in the past few decades and so I think we can innovate and note up our notions of zero with our newfound knowledge.
In my R&D, I explored how the zero used in Hindu-Arabic Notation plays at least two roles.
When it’s by itself on left side of the decimal point, it’s no objects; when it’s somewhere else, it’s presumed metadata about objects.
For instance, the zero in 10 represents a full stack of ten identity objects if decimal and a full stack of sixteen identity objects if hexadecimal.
But what if position notation is expressed as a data tree, represented by the horizontal and vertical planes, not just horizontal?
As a data tree, the zero in 10 can represent a container of various identity objects.
=> https://en.wikipedia.org/wiki/Stack_(abstract_data_type)
Thanks again, Kyle, for additional links. I too enjoyed The Book of Nothing.
You are obviously a computer person, so that is beyond me. Still, I very much see the value. Do you know Scott Arronson’s work in quantum computing and such? Funny that he’s now at UT in Austin where I live.
https://www.scottaaronson.com/blog/
I come at much of these issues via Turing, Godel and Cantor in a very basic way. If you have any insights into the intersection of math and visualization, I’d like to know… I was curious about Cantor’s triangles as a means of exploration, for example.
Thanks again for the interesting links and thinking.
Some minor corrections to the Q&A video:
– You wrote Z \otimes Z for the product of the group Z with itself, but that symbol is normally used for the tensor product, which in this case would be Z again. I think you meant to simply write \times.
– The symbol for the quaternions is not Q (that would be the rational numbers), but H, after Hamilton.
– The Z/2 action on S^3 is not given by reflection around the equator (that action is not free, and the quotient is not SO(3)), rather by the antipodal map.
By the way, thank you so much for these videos! As a mathematician with close to no knowledge of physics, I might not be the intended audience, but I’m enjoying these lectures immensely. I’ve learned a lot so far, and they’ve given me enough basic intuition and motivation to look into certain aspects of mathematical physics more seriously!
Thanks for the clear explanation of the intended scope of this series. It’ll help me to keep my questions relevant.
Hi William, that’s great you’ve read The Book of Nothing!
Thank you for letting me know about Scott Aaronson’s work. If there’s a particular post you really like, please let me know. Maybe email or tweet me though because I don’t want to clog up Sean’s blog with tangential correspondence.
My twitter is @bestape. To email, please go to http://newslettersignup.seedtreedb.com and sign up. You can unsubscribe after you sign up if you don’t want to receive my newsletters.
I don’t know quantum computing but I get the spirit of it, I think.
Perhaps what I call quantum logic is somewhat incorrect but it helps me figure things out, namely distinguishing determinate structure/process (classical) from indeterminate structure/process (quantum?). For instance, a determinate I/O is create/destroy and an indeterminate I/O is change/change-back, but whether change is a create or destroy change is unknown.
1/2 can represent this uncertainty. Which 1 of 2 is the 1 in 1/2 mapping to? Before I started expressing myself with JavaScript, I don’t think I would’ve cared about the ambiguity inherent in 1/2 but, after debugging enough subtle namespace collisions, I learned to care.
I also come at much of these issues via Turing, Godel and Cantor in a very basic way! What I gleaned from them is that an idealized diagonal has a perpetual motion machine attribution. So, Pythagoras and Zeno are part of the picture too.
I first started with continued fractions because I figured they were the simplest abstraction of nested functions in JavaScript, which was the phenomenon I was trying to deconstruct. However, I was confused about how continued fractions and diagonals were intimately related to each other.
It took me quite a bit of sleuthing to find Vera’s Metallic Mean Family, and then a bit more time playing around on Inkscape, before I figured out how a diagonal’s perpetual motion can be expressed as a parallelogram’s perpetual motion.
I think the USTPO is publishing my patent at the end of July, and the non-provisional filing includes most of my field notes on the subject, so look out for that if you can (or sign up to my newsletter and I’ll announce its publication).
In the meantime, I’ve posted some of my field notes on Wolfram at http://translate.powerlawgeometry.com and on YouTube at http://playlist.diagonalmethods.com
This Instagram post integrates many of my ideas into a single visualization: https://www.instagram.com/p/BzeLOHsgXwh/
All the best!
Thanks again Sean for this amazing series!