Deriving the Dirac Equation
Вставка
- Опубліковано 8 тра 2024
- In this video, we'll derive the Dirac equation, and see where it comes from! :)
Recommended reading: Introduction to Elementary Particles, by David Griffiths, Chapter 7.
Equations from the videos are available as downloadable PDFs on my Patreon. I'll also be on there to answer any questions you might have.
/ richardbehiel
Chapters:
0:00 Intro
0:38 Three Principles for the Dirac Equation
3:12 Square Root of the Mass Shell
7:30 Anticommutation Relations
9:50 The Dirac Matrices
10:58 The Dirac Equation
13:58 Spinors
#physics #quantum #relativity
Hey all, thanks for checking out this video! :)
While reviewing the video just now, I realized I probably should have been more clear about covariance and contravariance. This affects the sign of the momentum terms (more generally, the space-like terms), and is a perennial source of dropped minus signs 😅 The plus signs at 13:20 are due to using the contravariant Dirac matrices, three of which which absorb the minus signs, see the Wikipedia article “gamma matrices” for more info.
Anyway, the main idea I hope you’ll take away from this video is that the anticommutation relations for the Dirac matrices arise from taking the square root of the mass shell.
I guess it'd be better to pronounce that plus sign before mc at 12:44 is due to conjugation of an i at the denominator.
ok, so are spinors, 720 degrees to return to its "start", due to, 360 degrees of spin in the electric and then 360 degrees of spin in the magnetic fields, which would then return to its start?
The 1st order space-time variables as you pointed as breathing motion is non commutative of matrices makes space-time an illusion. And the spinors as real .
Is it ?
What do the Twistors of Roger Penrose and the Hopf Fibrations of Eric Weinstein and the "Belt Trick" of Paul Dirac have in common?
It takes two complete turns to get down the "rabbit hole" (Alpha Funnel 3D--->4D) to produce one twist cycle (Quantum unit).
Can both Matter and Energy be described as "Quanta" of Spatial Curvature? (A string is revealed to be a twisted cord when viewed up close.) Mass= 1/Length
Dirac was known for both the brevity and precision of his speech. At a conference he was lecturing and writing at a blackboard. A member of the audience said at the conclusion of Dirac's presentation, "I don't understand your equation in the upper right." Dirac said nothing and returned to his seat at the table with other lecturers. After an uncomfortable silence, the conference moderator asked, "Mr. Dirac, are you going to answer the man's question?" Dirac: "It wasn't a question. It was a statement."
Classic Dirac 😂
Yes, he was a special man. Meeting Werner Heisenberg for the first time, his first question was: "Do you have an equation too?"
@@veronicanoordzee6440wasn’t that Feynman?
@@sleepycritical6950 Yes, could well be.
@@sleepycritical6950 No-no, Dirac proper. Feynman wasn't so pedantic, he played drums, picked up girls... and all that...
The "I'm letting the squares breathe..." was genius for geometrically visualizing variables!
I'm a maths teacher, I've hand waved this idea so many times. I wish I could make animations like this XD
Well, I find breathing squares neat but don't quite get what this breathing is intended to convey...
It’s just there so we’re not tempted to try to pick a specific arrangement of aspect ratios, then try to cancel two rectangles with one rectangle, for example. The breathing (variability) shows that each rectangle has to cancel out with its own symmetric counterpart, since that’s the only way the solution will hold in general.
A thousand jargon-filled wikipedia articles could not have given me the clarity I now posess thanks to this video. Thanks so much
The reason the coefficients don't commute (so the opposite breathing squares cancel out) and why the coefficient squared are negative numbers, is because a spinor is a unit quaternion! In fact, the nature of spin is a bit less mysterious if instead of using Dirac matrices we use quaternions, for rotations on a 4D hypersurface :)
You beat me to it by 5 hours :)
When I saw the constraints I just jumped up from my chair and shouted "quaternions!". My head is now spinning, trying to decide the difference between physicists and mathematicians...
even better, a Real 3D Clifford Algebra, where the BiVectors e1e2 = i, e2e3 = -j, and e3e1 = k
All spinors are rotors, and all rotors are double covers of their respective rotational range.
It's much less mysterious then, because quaternions are known for how easy they are to understand.
@@TheBlindfischLP the basics of quaternions are not hard. Their more advanced uses get a tad bit more complicated, but so do complex numbers. The important thing is that once you use quaternions you know you're dealing with a rotation.
When the math is shouting “PARTICLES AREN’T FUNDAMENTAL, THEY’RE EMERGENT, YOU’RE JUST DOING EPICYCLES AGAIN”
Straight-up the best physics videos on youtube
At least the best on Dirac and mentioning spinors!
The teacher I had for QFT was one of the best teachers i ever met, and yet, this is so much more insightful then what he ever managed on a blackboard. The visualisation is genius.
Wow this lesson is much better than in Oxford or Harvard. Very interesting to see visualisation of solution of Dirac equation to compare with Klein-Gordon and Schrodinger equations. Also very interesting to see how do the wave functions in the bispinor responsible for spin up, spin down, and antimatter affect each other
Fun fact. Back in 1972 I was taught some mathematical physics by the late Joe Moyal. Joe was invited by Dirac to Cambridge during the World War 2 to discuss Joe's work on statistical foundatiions of quantim mechanics (there is a 1947 paper with Bartlett and Kendall which also deals with the issues). Joe met Dirac and I can say that I have shaken hands with someone who has shaken hands with Dirac.
Very cool! I’m jealous!
Wow. I have devoted all of my time to getting into GR and have only had a basic rundown of QM but have wanted to get into the more interesting QM stuff for a while. You did a great job and I love hearing your excitement for introducing new ideas. That was perfect and felt so clean. Great stuff!!!!
Thanks for the kind comment, and I’m glad you enjoyed the video! :)
eigenchris's "Spinors for Beginners" series is a really good introduction to, well, spinors for beginners!
Absolutely terrific :)
btw, here's a tip for everyone who sees this: geometric (Clifford) algebra really helps understanding these spinors better (and also space-time in general).
the spatial spinors are quaternions, or the 3d real geometric algebra where e1e2 = i, e2e3 = -j, and e3e1 = k. The time-spinor gamma_0 is more mysterious, as it is not the trivector e1e2e3, because that squares to -1 also, not +1.
I second this! In particular, the gamma matrices have a one to one correspondence with the basis vectors of spacetime algebra (signature Cl(1, 3)), which are also often written as γ0, γ1, γ2 and γ3, making the conversion even more convenient!
I recently started my postgraduate physics studies and one of my modules (which is by far the hardest one for me) covers the Klein-Gordon equation and (now) Dirac equation. The reading material and textbooks for most of the part pretty much assumed the reader knows how to derive certain things and/or where it comes it or what it represents etc, which is why I'm finding them incredibly difficult. Your videos have literally saved my life by providing the much needed intuition for these topics. Keep up the amazing work!
I’m glad to hear that! :)
I ve began studing physics for past 5 months from ground up, and this one got me for reall this time difficult bad boy
Thanks for such a clear explanation. Can you also elaborate on why A,B,C,D matrices should be 4x4. Like what should be motivation behind that to choose 4x4 instead of any higher like 16x16 or so?
I never heard of this way of deriving the equation and now the matrices seem so much more understandable... Congrats dude, you just did the impossible of explaining all of this stuff I tried to understand for months
That’s the best possible comment I could get on a video. Thank you! :)
You should read “quantum field theory for the gifted amateur”, it’s a texture newish book, but it has a great way of teaching these concepts, they also start with first deriving the KG equation and then introducing “taking the square root of a differential operator”
Thank you so much! I was reading the book “The strangest man” by Graham Farmelo, and your video gives an in depth looking into Dirac equation. This let me give the following parallel from a basic math concept to understand Dirac equation.
While Cartesian coordinates, introduced by René Descartes, provide a structured way to map points in space using two or more axes, the Dirac matrices, part of the Dirac equation in quantum mechanics, offer a profound mathematical framework for describing the behavior of particles in spacetime.
Just as Cartesian coordinates allow us to pinpoint locations in space with precision, the Dirac matrices enable us to describe fundamental properties of particles, such as spin and angular momentum, within the framework of quantum mechanics. Both systems provide a means to understand and navigate complex phenomena: Cartesian coordinates in the realm of classical mechanics and geometry, and Dirac matrices in the intricate domain of quantum mechanics.
In essence, while Descartes paved the way for understanding space in terms of ordered pairs of numbers, Dirac expanded this notion to encompass the intricate fabric of spacetime at the quantum level, offering a beautifully intertwined perspective on the mathematical underpinnings of the universe.
Good comment.
I was watching Alexander Fufaev video on the Schrodinger equation (He gave an excellent explanation), and he mentioned Diracs equation as the next thing for relativistic SE.
I'm happy to have found this. It's been years since I finished.
So here I am learning new things and brushing up the old, to enable drawing out something new if I persist just enough.
This is excellent and widened my understanding but raised further questions too.
Excellent presentation. I've seen other videos on the Dirac equation but this is the first one that makes me feel like I (just about) understand it!
It's also worth noting that the matrix approach is entirely equivalent, mathematically, to the other ways. It's not just that matrices happen to solve this particular problem - matrices form a group, and that group is isomorphic with any group that obeys the same multiplication rules. So not only is this "a way to do it" - you also don't lose ANYTHING by doing it this way.
You’re right to point out the isomorphism between the various approaches, and I agree.
I also use the matrix approach much, much more than the STA approach, because it lends itself so well to python codes, as well as just to the kind of math I’m familiar with. And it’s much easier to check your work when you’re using the same language as most of the textbooks on the subject.
But in defense of STA, I do think it’s a slightly more direct and elegant way to visualize the math. It appeals more to the intuition, but IMO is harder to actually use (that’s probably my fault since I’m not great at geometric algebra).
At the end of the day, it’s helpful to consider various perspectives, even if they’re isomorphic. Studying the different points of view can make a lot of things click, that otherwise wouldn’t have. Honestly I didn’t really understand what the Dirac matrices were, until learning STA. They seemed very mysterious to me, even though I understood the algebra. But the STA approach taught me not to over-mystify these, since they’re “just” the basis vectors for CL1,3(R). I missed out on that insight for many years, even though I had been taught the Dirac equation and had been using it in various simulations.
en.m.wikipedia.org/wiki/Spacetime_algebra
I think this is the clearest picture of Dirac equation and I have minimal training in math and physics 😊
so true, im just at my first semester and i was able to understand most of the things, this man should be proud of himself
Brilliant content! Looking forward to the next installment on spinors!
This is an excellent video! It does a great job making the solution feel intuitive!
I know the Dirac equation but never really saw it being derived before. Either I was incredibly blind in my education or my educators hand waved everything and I struggled from there. These videos are opening my eyes more than my entire journey through undergrad and honors.
This sort of accessible education will prove to be revolutionary.
Let’s hope so! :)
Yeah I did physics and so I had seen the reasoning he describes in the video. However, watching it, I crave going back and reading those notes now. At the time they seemed so insane. Now I think it would be a joy to read and I think all people going into physics in the future are going to be so grateful for the existance of videos like this :)
You do a really nice job explaining these difficult concepts both formally and casually. And your Manim skills are impressive!
This is fantastic. A secret closely guarded by academics has been simplified so that even I can understand it with only a BS in Engineering. Thank you
I need to watch more on spin as it too is a kept secret usually presented and discussed in baloney classical ways. Just a glimpse here helped already.
I would love for someone to simplify Hilbert space as it too is slung around a lot without explanation.
Fantastic explanation! Loved the visualization of the breathing matrix and the colour coding of the terms to make it immediately obvious that the coefficients don't commute and the need for Dirac matrices. And then adding the flag to illustrate the double rotation.. Fantastic! Thank you for your time effort on this video
Thanks for your kind comment! I’m glad you enjoyed the video :)
So beautiful. You didn't even get to the half of the video and already all the mystery unveiled. Why nobody told me about this when I was studying theoretical physics is beyond me. It is both obvious and simple, and derivation is logical and could not be otherwise. To me the commutation relations were always introduced without any explanation or big reason other than this makes thing work, but not really. All the implications later are just simple logical conclusions (they might be hard to derive, but there is nothing mystical about them).
You are a treasure! You just put so much into perspective for me wrt spin states, what the equations represent, how they come about. Great video! Thanks
Glad to hear that, thanks for watching! :)
Been waiting for this one since your last video! Awesome work!
Thanks Raspberry! :)
Thanks for clearing everything up!❤
was a very nice video btw :). and the quaternion modification, i just figured it is easier to call the non commutative root of plus 1 "u" u*u=1 , u*i=i i*u=-i ect.. then it works out, cleaner than special rules for minus one or one, where you have to basically write the same thing anyway.
I said out loud "Ooh!" excitedly when i saw this video as i was working on visualizing the Schrödinger equation for the Hydrogen atom!
The requirement to complete 2 360 degree rotations in order to reset reminds me of mobius strips.. I'll need to watch this lovely video a few times. Subscribed.
Just thank you, as clear as it can be
Okay , so I thought Dirac already knew about spinors before combing SR and QM. Thanks to you , now I understand that he might have went for them after finding these relations of A , B , C and D.
That's nice because sometimes I feel like I should go for a masters in mathematics rather than physics. I am gonna go and be happy with physics for sure now because of this video.
Great explanation of complex subject.
So beautiful tears came to my eyes. Please update to a framework based on Clifford Algebra or Geometric Algebra rather than the most commonly used framework. Thanks for your service!
I love these videos! Great explanation and visualization!
Thanks, glad you enjoyed the video! :)
The way the commutation relations came out from x^2 = E^2-p^2 was so cute. Also the anticommutativity is very reminiscent of geometric algebra and specifically space time algebra.
If you haven’t read it yet, you might enjoy Hestenes’s papers on the Dirac equation in STA. In my opinion it’s the most elegant way of formulating the equation. Hasn’t totally caught on in the mainstream physics community, but it seems to be gaining popularity over time. Good ideas always win in the end.
@@RichBehiel could you give specific links to the papers?
@gaHuJIa_Macmep yes, let me look for those later this evening. I’m about to get on a boat for a day of scuba diving, but I’ll follow up when I get back. Please remind me if I forget!
@@RichBehiel Oh! Scuba diving! Where are you situated then? Somewhere in southern hemisphere? It's winter in the northern one now...
@gaHuJIa_Macmep well I live in Santa Barbara, but I’m currently on vacation in Hawaii! :) Went diving today near Honolulu, just east of Pearl Harbor. Saw a bunch of huge sea turtles. Good times.
I found the squares (and the whole derivation) breathtaking
I'm an undergrad physics student and you made this look incredibly easy. I'm sure I don't really understand the geometry of it but I have a way to imagine the dirac equation now, thank you.
Glad to hear that! :)
Fantastic! Looking forward to spinors video :)
A spinor is best thought of as a Rank ½ Tensor.
Well, Diract went a different direction. When I saw those constraints, my brain immediately went "quaternion"
I'm not understanding how this system of equations could be solved with quaternions. Perhaps I'm missing something, but I don't think the topology or geometry of the quaternions is the same as that of the space needed to solve this system of equations.
You need 4 numbers, all of which are pairwise anticommutative, where one of them squares to 1, and the other 3 square to -1. In the quaternions, we can certainly get 3 of these - for instance, i j and k are all pairwise anticommutative, and all square to -1. However, I can't figure out what the 4th one would be (the one with the opposite sign when squared).
1 and -1 both square to 1, as needed, but these both commute with all of i, j and k, and everything in the quaternions (because they are central elements of the quaternions, and really all number systems). These are the only square roots of 1 in the quaternions, because the factorisation x^2 - 1 = (x-1)(x+1) = 0 is true in the quaternions (it is a division ring, where -1 and 1 are central). So there is no square root of 1 which is anticommutative with anything, which it needs to be for this system of equations.
@@stanleydodds9 The Dirac equation describes BISPINORS. These are isomorphic to biquaternions, also called complex quaternions. You can do Weyl decomposition to get spinors, which are isomorphic to quaternions. So yes, technically you can't use quaternions in the Dirac equation, because they only are "half" of the solution. If you look at the gamma matrices you will see that they contain the Pauli matrices, which are unit quaternions, and some of them are multiplied by i (=complex quaternions) :)
@@FunkyDexter of course, yes, some degree 2 extension of the quaternions is going to have enough freedom to get this extra number. And naturally it would turn out to be the octonians, given that it's the "nicest" (or at least most interesting/useful) 8 dimensional algebra, in some sense.
Just saying that it's quite clear that purely in the quaternions themselves, it's not possible to satisfy this system of equations. And of course it's related to the fact that quaternions rotations don't form a double cover.
@@stanleydodds9 unit quaternions do form a double cover to real rotations. It's the whole reason they are interesting in the context of spin, and why they can be used in computer graphics. The jump to octonions is useful in the context of a unified description of the wavefunction for both the electron and the positron, but it has more to do with the inherent chirality of spinors rather than extra rotational degrees of freedom.
God tier explanation, can’t wait for more stuff
Thank you for another great video :) Cant wait to learn more about the topic.
Thanks for watching! :)
Informative and concise, great video 🤍
Thanks, I’m glad you enjoyed the video! :)
I'm astonished. This is beatiful
Wow!! Really amazing explanation! Thankyou very much
Thanks, I’m glad you enjoyed the video! :)
Wow though I have just studied 1 year of math at the college level I understood the gist of this well narrated video. Good job!
Thanks, I’m glad you enjoyed the video! :)
Keep up the good work.
This was awfully well timed, we just saw Dirac's equation last week in our quantum class 😅
At the time 12:13 , this equation can only hold as an operator because left side is a matrix and right side is a constant. This means that actually what we are find is only can be satisfied in quantum mechanics and does not any analog representation in classical mechanics.
Perfect, thanks!
Once again, THANK YOU!
Thanks for watching! :)
great explanation and diagrams!
Thanks! :)
I have zero background in physics and was just listening out of curiosity. But I know linear algebra pretty well and when you said "non-commutative multiplication" I yelled, "they're matrices!" Ha!
VERY cool. Thanks for this!
Thanks, I’m glad you enjoyed the video! :)
Brilliant!
Simply a great video !
Thanks, I’m glad you enjoyed it! :)
Richard: 10:39
My physics prof: I'm gonna pretend i didn't hear that.
😂
Good work
man,this equation is a kind of awesome awesome art! paul dirac is a genius and one math artist! awesome!
Dirac was enamored by the beauty of math and physics. You can see why!
brilliant !
Wonderful!!
First, excellent video. Second, one technical point at 14:50 or so, Psi 3 actually corresponds to spin DOWN positron and Psi 4 to spin UP.
Thanks! :)
I’m confused, but I definitely appreciate your feedback! In “Introduction to Elementary Particles” by Griffiths, equation 7.30 and the following sentence state that psi3 is spin up positron, and psi4 is spin down. But then later he converts the u3 and u4 spinors into v2 and v1, respectively, such that now u1 and u2 are spin up and down electrons respectively, while v1 and v2 are spin down and up positrons respectively. So the “up” and “down” are reversed when looking at the indices on the u’s and v’s, but when writing the bispinor as a column, it should still go up, down, up, down, right? Or am I missing something?
This is beautiful.
i leave it as an exercise for the content creator to look at the differences between the gamma matrices and these types of quaternions with a non commutative real unit.
That was surprisingly quick. And you have indeed done a decent job ;).
Thanks, I’m glad to hear that! :)
I loved it! Awesome!
Glad you enjoyed it! :)
Anche qua la Fisica ,( ci sono le numeriche o variabili della Fisica ,No!)non c'entra nulla , è una ricerca che è già stata esposta, dalla Dirac Converse , riguardo gli andamenti del ciclo di vita del prodotto,nelle sue varie fasi, durante una linea di produttività. Ciao ciccino un bacione! Può servire la successione di equazione per il riscontro di un andamento o processo di distanziamento. Questo si è vero! Bene.
Soooo superbly made😍😍
Thanks! :)
Amazing video! Thanks!
Thanks for watching! :)
This is completely ad-hoc reasoning that it is amazing that it works.
The physicist’s approach to mathematics! 😂
if you modify the quaternions slightly, that is you fuck around with 1 and find out, then its all good as well. what you need is simply to modify the real unit multiplication such that if you multiply the real unit by a non real unit A=+-1 B=i,j,k such that if A*B, A=1 and if B*A, A=-1, then it satisfies the same constraints, basically you make 1 non commutative and you are good to go.
Such a good video. I only wish it was formulated in terms of geometric algebra. One of the major reasons spinors are trippy is because they're usually all about exact coordinates, while for the purposes of explanation it's possible to split it into two parts: "thing where squares are -1, and AB = -BA, and their algebra" and "how to represent numbers of that algebra as 4x4 matrices to do an actual simulation on a computer". You can't neither draw nor get geometric meaning out of numbers of 3D spacetime algebra anyway.
We’ll get into Hestenes at some point :)
When you hit the bit with the anti-commutativity and the squaring of A,B,C,D to 1,-1,-1,-1, I thought "Hey, isn't that a rotor?". A would be the scalar part, and B,C,D the bivector parts.
Great job dude
Thanks! :)
Yes, the pleasure of dealing with "highly non-trivial objects"!
it's here! hurrah
🙂🙂 , Complimenti!
Thank you!
Thanks for watching! :)
Great explanation, and helpful visualizations! However, I think a motivation for the third principle of the Dirac equation would have been nice. Now it was just stated, for no obvious reason. (I know the motivation if it myself since before, but other people may not know it.)
A thing that I'm wondering-at 7:31 when you write out the equations for A, B, C and D, this strikes me as equivalent to the equations for the basis elements of the quaternions, so would it be possible to write the Dirac equation with the quaterion basis elements instead of the gamma matrices, and letting the wave function be quaterion-valued instead of a vector field?
basically if you replace 1 in the quaternions by a non commutative root of 1, that is what i said by a multiplication rule that picks out either 1 or -1 then we are all good.
Really good. Now if we could just come up with a way to simplify this by cutting out two of the three spatial dimensions.
Amazing video! I do have a question about the final form of the equation. Since you are multiplying bispinors by 4x4 matrices, does that mean that the “=0” in the dirac equation is the “zero bispinor” or is it literally the number 0?
Great question! Yes, it’s the zero bispinor. One way of thinking about the Dirac equation is that it’s four coupled differential equations, and the Dirac matrices intertwine the various components of the four-momentum across those equations, and each equation has a zero on the right side.
Beautiful
Amazing explanation and easy to follow, something that I was not expecting when considering how to derive the great Dirac equation. Question (I'm not familiar with Klein-Gordon equation): what is the motivation to get a first-order equation? A second-order equation incidentally does not predict spin?
Good question! :) I talk about this in the second half of my KG video. Long story short, the second-order makes probability densities and currents weird, such that sometimes you have negative probability density. This has been contextualized over time, as QFT has developed, but back in the day it was one of Dirac’s main motivations to look for a first-order equation.
@@RichBehiel Thanks! I'll check that video as well!
The constraint of 1st order space and time dependence eventually lead to the Dirac field/"wavefunction" being bispinors, signifying spin-1/2 (fermion). Is it possible to come up with an equation 1st order in space and time, for integer spin fields (bosons)? Is the Klein-Gordon equation the most "elementary" equation for bosons?
even the way you have the visuals in motion = NB ... ty
that 4x4 matrix breathing is important for understanding reality.
@6:55 lol case in point
You did good!
Thanks! :)
I like my least paths solution to causality: e^(M*{L/lambda}^2)=cos(Lambda*M)+lambda*sin(Lambda*M)!!!, "The Temple Equation for Causality."
Far out Man, that 'was' real trippy! :-)
Thank you
You’re welcome, thanks for watching! :)
Very nice. I almost understood this, but I don't get what the deal is with the minus sign of the momentum operator. It's usually in QM p_x = - i \hbar \partial_x .
But you used initially p_x = i \hbar \partial_x with the plus sign, but write in the same line in the beginning at time stamp 1:35 \hat{p} = -i \hbar (\partial_x, \partial_y, \partial_z). Then, in the end, in the Dirac equation it's always p_u = i \hbar \partial_u . Is the trick somehow that the imaginary i should be explained from taking the square root of the metric tensor, or signature (+---) or (1,-1,-1,-1) which makes (1,i,i,i) ? I dunno. This is a bit confusing to me right now, I must have missed something.
Very good exposition. I was surprised I could follow it as well as I did. It even allows one to play with the matrices to confirm the results for oneself.
I never understood it before.
The Wikipedia exposition, by contrast, I find very difficult to follow.
Thanks, I’m glad you enjoyed the video! :)
Great video! Spinors make my head spin. I really can't grasp why these weird, quirky mathematical object describes the universe so well.
The equation appearing at 1:45 equates a left hand side which is a matrix to a right hand side which is a scalar. Is that meaning that we are supposed to understand that this scalar is multiplying an implied unit matrix ? Same for all the following equations ?
Is it possible to "Dirac-ify" other differential equations in physics? The wave equation is to the second power; are there any mathematical objects we can use to make the wave equation first-order in time?