- EIGHTH MEETING
Come on down to the EIGHTH QUANTUM MEETUP! We'll be continuing our discussion of the theory of constellations aka the representation theory of SU(2) in disguise. We'll be considering ways in which the Majorana representation allows us to appreciate certain amazing dualities: between spin-j states and permutation symmetric states of 2j qubits, and between spin-j states and states of the two dimensional quantum harmonic oscillator. As usual, food and visualizations will be provided! If we have time, we'll get to particle interactions--collisions and splittings--that conserve angular momentum. Hope to see you there!
- SEVENTH MEETING
Let me know if this date works for people! In our seventh meeting, by popular demand, I'll try to go more into the specifics of the Majorana Stellar Representation on the whiteboard, etc. I've compiled a list of papers for reference of varying degrees of sophistication and detail: Viewpoint: A Quantum Constellation https://physics.aps.org/articles/v5/65 Stellar representation of quantum dynamics http://www.esrf.eu/news/spotlight/spotlight163/index_html Geometric theory of quantum spin systems https://indico.in2p3.fr/event/8089/contributions/43803/attachments/35190/43407/Berry_Majorana_CPTG_Annecy_P_Bruno.pdf GEOMETRY OF PURE STATES OF N SPIN-J STYSTEM https://wwwold.fizyka.umk.pl/~kolenderski/viewer.php?file=Posters/smp42.pdf Quantum geometric phase in Majorana's stellar representation: Mapping onto a many-body Aharonov-Bohm phase https://arxiv.org/abs/1204.2372 Representation of Berry phase by the trajectories of Majorana stars https://arxiv.org/abs/1406.6821 The Exact Curve Equation for Majorana Stars https://www.nature.com/articles/s41598-017-15776-w MAJORANA REPRESENTATION OF HIGHER SPIN STATES http://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Angular%20Momentum,%20Spin/D2.%20Majorana.pdf Extremal quantum states and their Majorana constellations https://arxiv.org/abs/1503.03446 Barycentric measure of quantum entanglement https://arxiv.org/abs/1112.0437 Majorana representation of symmetric multiqubit states https://arxiv.org/abs/1103.3640 Classification of Entanglement in Symmetric States https://arxiv.org/abs/1110.5200 Symmetric entanglement classes for n qubits https://arxiv.org/abs/1103.0271 Geometric Entanglement of Symmetric States and the Majorana Representation https://arxiv.org/abs/1010.4777 The maximally entangled symmetric state in terms of the geometric measure https://arxiv.org/abs/1003.5643 Entanglement and symmetry in permutation symmetric states https://arxiv.org/abs/1001.0343 Entanglement in the symmetric sector of n qubits https://arxiv.org/abs/1101.2828 N-qubit states as points on the Bloch sphere https://arxiv.org/abs/0910.0630 Geometry of spin coherent states https://arxiv.org/abs/1710.11326 "Anticoherent” Spin States via the Majorana Representation http://www.ejtp.info/articles/ejtpv3i10p143.pdf Anticoherence of spin states with point group symmetries https://arxiv.org/abs/1509.08300 Coherent-State Approach for Majorana representation https://arxiv.org/abs/1601.07612 GEOMETRY OF QUANTUM STATES https://pdfs.semanticscholar.org/3f28/893b7e8c5c96525493db8e3d6b09ab47f426.pdf ROAD TO REALITY http://chaosbook.org/library/Penr04.pdf Majorana Representation in Quantum Optics: SU(2) Interferometry and Uncertainty Relations http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A1091994&dswid=-9849 Geometric description of modular and weak values in discrete quantum systems using the Majorana representation https://arxiv.org/abs/1612.07023 Generalized Weyl-Heisenberg algebra, qudit systems and entanglement measure of symmetric states via spin coherent states https://arxiv.org/abs/1804.06184 Composite Pulses in N-level Systems with SU(2) Symmetry and their Geometrical Representation on the Majorana Sphere https://arxiv.org/abs/1708.06842 The Majorana representation of spins and the relation between SU(∞) and SDiff(S2) https://arxiv.org/abs/hep-th/0405004 Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere https://arxiv.org/abs/math-ph/0408046 Correlations between Maxwell's multipoles for gaussian random functions on the sphere https://arxiv.org/abs/math-ph/0410004 Bloch sphere representation of three-vertex geometric phases https://arxiv.org/abs/1107.5447 Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates https://arxiv.org/abs/1703.06102 Complex polynomial roots toy https://duetosymmetry.com/tool/polynomial-roots-toy/ Hope to see you there!
- SIXTH MEETING
Hey everybody! I know this isn't much of a heads up, but if you're available this Saturday, come on down to our SIXTH QUANTUM MEETUP. (We could also meet Sunday instead, if people are interested in that--let me know!) Ideally, we'll be discussing a number of topics. First: the polarization of a photon. I have some real life linear and circular polarizing filters that we can play around with and put in sequences (as far as I know, this is the cheapest "quantum" experiment money can buy!). We'll discuss how the "ellipse" traced out by a propagating light wave corresponds (mathematically and physically, in a beautiful way) to a qubit! It turns out we can apply much of what we know about spin systems and Stern-Gerlach apparatuses to photons and polarizing filters: the sphere, as usual, comes to our rescue. I have some 3D graphics to demonstrate. Then, time allowing, I'd like to discuss some foundational issues in mathematics. I want to give a sense of how quantum mechanics arises in a natural and organic way from the concept of "number" itself, stretching back to the Ancient Babylonians and earlier. We'll see how due to the simple demand of consistency, the concept of number evolved from something you can count like a pebble and put in a pile, to something you can measure out with a string, or enclose with a fence, to something which can convert one currency to another, or sound against another to create harmony or dissonance in accordance with its primes. But then numbers like the square root of 2 were discovered to be irrational and -1 to have no square root at all? We need to move to 2D: to be consistent with algebra, numbers turn out to have to be right triangles, aka points on a plane. Multiplication now corresponds to a stretch by a hypotenuse and a rotation by an angle. Furthermore: if you fold up the 2D plane into a sphere, closing it at the north pole by adding an extra "point at infinity," we might say a number is: a point on the sphere aka a qubit. (At this juncture, we'll return to the polarization ellipse and consider it anew.) But is there a next step? We'll begin to explore how the amazing answer to that is just: more points on the sphere!--thanks to the fundamental theorem of algebra. (To wit: there's a progression from: natural numbers -> integers -> rationals -> complex numbers aka monomials aka qubits -> polynomials. Then miraculously, polynomials turn out to be vectors and matrices in disguise--and we have linear algebra! (Bonus: Continuing the story, since all finite groups have linear representations, this takes us pretty far up the hierarchy of abstraction, all within the sphere. Also, lots of infinite groups have linear representations: like the symmetry group of the sphere itself, for example. Then, there are the nonlinear infinite groups...) Anyways, hope to see you there!
- FOURTH MEETING
After a long hiatus, I welcome you all to a fourth Brooklyn Quantum Meetup! We'll be discussing the interconnections between algebra, perspectival geometry, astronomy, and quantum physics. We'll be trying to imagine the night sky from the perspective of an electron. Fixing an observer here and now choses a slice of Minkowski space: all the intersecting worldlines can represented as points on the observer's celestial sphere. Lorentz/Mobius transformations on this celestial sphere represent motion through spacetime: the stars seen from a different point of view. Coincidentally, n-dimensional quantum states can also be represented as points on a 2-sphere, n-1 points specifically. This is called Majorana's stellar representation of the quantum state. Empirically, if you orient your Stern-Gerlach apparatus in the direction of a Majorana star, and send your state through the apparatus, one of the resulting spin states has 0 probability. Thus we can interpret applying Lorentz/Mobius transformations to the Majorana's stars on the sphere as correctly giving us the relativistic transformations of the quantum spin state. How does it work? To each quantum state is associated a complex polynomial, whose complex roots are stereographically projected from the plane onto the sphere, where "poles" of the function (where it goes to infinity in any direction in the plane) are all mapped to the north pole of the sphere. It works because a) the fundamental theorem of algebra: any polynomial equation of degree n has n roots in the complex numbers, and so any equation can be factored into 2d points b) the identification of all poles out in any 2d direction with a single "point at infinity" which is joined to the complex plane. Picture wrapping up the plane into a sphere, you have to add a navel, a bellybutton, where the surface comes together at the north pole. That's the point at infinity. Think of the point at infinity like the point where the railroad tracks meet in the distance. It's a perspectival point, and we've created a perspectival space where poles and roots can be represented on equal footing, and in which we can formulate quantum mechanics and relativity both. We'll get to play around with some python code I've written that'll help us visualize and interact with the "3d geometry" of quantum spin. We can project it up on the wall! In other news, I've started a blog that if you've enjoyed this meetup, you'll probably enjoy as well. You can find it at http://timespaceinterchange.com.
- THIRD MEETING
Hey all! So for our third meeting, we'll be discussing free will and quantum teleportation! You can peruse John Conway and Simon Kochen's papers on free will here: The Free Will Theorem, Conway and Kochen (2006) https://arxiv.org/pdf/quant-ph/0604079.pdf The Strong Free Will Theorem, Conway and Kochen (2008) https://arxiv.org/pdf/0807.3286.pdf Don't worry about trying to follow the whole argument. If you've never read mathematics papers before, you should know: the first time you read a paper, don't try to follow every step. The most important thing is get the whole shape of the argument, from start to finish, into your head first. Once you know where you're going, it's much easier to understand where you start. Scattered about are some more philosophical statements--focus on those. Furthermore, Conway gave a lecture series on the theorem, which I highly recommend. Here's a link to the first lecture: https://www.youtube.com/watch?v=ftIllWczf5w&t=967s To save you time, someone pulled out his final concluding statement in the last lecture: https://www.youtube.com/watch?v=E7-LX1tF3iw I quote: "You know I feel I ought to say one thing. It's okay to believe in free will. There's no compunction to believe in determinism. ... There's no reason to believe in determinism whatsoever. There's never been any evidence for it that hasn't disappeared with the development of quantum mechanics. And it's the natural thing to believe. Seems to be true. If you disbelieve it, you ought to at least come up with some evidence. I don't believe anybody ever has." In other words, today the burden of proof lies upon the determinist to prove determinism. Determinism is the radical position; free will is the conservative position. I think this is the most important, but least appreciated, scientific fact of our time. Also, I want you to note the tone of voice in which he says: "It's okay to believe in free will"--as if calming down a stressed child. Putting the mathematics aside, one thing I am very interested in is: why might it be difficult for people to accept that we have free will, even if they understand, at least somewhat, the arguments in its favor. This treads on psychological territory, and because we all have our pre-existing beliefs on the subject, I want to warn you that, in the language of trauma theory, you may be "triggered" by our discussion. I'll do my best to make this a safe space. Not that we have to talk about our personal lives, but it might be interesting to think about where, when, and why we initially developed our beliefs on the subject of freedom vs. determinism, what kind of decisions we've made in our lives because of those beliefs, and what were the consequences of those decisions. This may seem beside the point, but in my opinion: Quantum mechanics only seems abstruse and weird if you approach it as a deterministic theory of reality. In contrast, quantum mechanics is almost intuitive if you approach it as a theory of how nature balances our own freedom against the freedom of everything else. Furthermore, at least psychologically speaking, a synonym for determinism is: slavery, submission. Therefore we find ourselves in the disconcerting situation where your prior philosophical beliefs, and your emotional attachments to them, may be hindering your understanding of the mathematics. I myself still don't know the best way to handle this, and so I invite your help. I was trying to decide on what readings to give y'all about quantum teleportation. In the end, I think they all require too many prerequisites, and since we're each at different levels of mathematical sophistication, I think it would make more sense to read over the wikipedia article: https://en.wikipedia.org/wiki/Quantum_teleportation and then come up with some questions. I'll start the meeting by showing you how to represent quantum teleportation on the whiteboard in the diagram language of Coecke et al, and then we can look back at the wikipedia article, and I'll point how how the different features of the diagram correspond to the "Hilbert space formalism" used throughout. This will give us a feel for how to think about quantum processes, and from there we'll naturally move into a discussion of free will. To whet your appetite, here's a little preview: There's a theorem proved a few years ago by the great John Conway and the great Simon Kochen called the Free Will Theorem. It starts off by assuming some basic stuff that's been demonstrated to everyone's satisfaction: like things can't travel faster than the speed of light, interactions take place "locally," things can get entangled through their interactions so that if later you ask them the same questions, they always give the same answers (or their answers are correlated in more general ways), stuff like that--and the theorem proves that: if we have the free will to choose what questions to ask nature, nature has exactly as much free will to choose her/his/its answers. Nature respects this symmetry whereby everything gets to be as free as possible. I won't describe here how he proves this, which is elegant, but mathy--but instead simply interpret the world in light of it for you. To take a classic physics example: an electron. You can ask an electron about its "spin." But there are three different ways to do this. You can measure its spin in either the X Y or Z directions, and that's your free choice. Symmetrically, the electron gets to freely choose to answer: UP or DOWN in the direction you chose to ask about. You freely ask a question, you get a free answer. You, freely choosing what to ask, experience yourself as free, and might say that the electron's spin is "governed" by "probabilities"--for example, 50% : 50%!--but conversely, the electron--if it had a brain or whatever--would consider itself free, and you governed by probabilities. That is, assuming you and the electron try this experiment a bunch of times: obviously a single experiment can't give you a probability! If things get entangled through their interactions, their answers to certain questions start to "rhyme" in various ways. But what is entanglement? The answer: it's just you freely asking me a question and getting my free answer from the perspective of a third person who isn't in the room with us. Our Q & A would have to be described by this third person as you becoming "entangled" with me, precisely because this third person can't know what question you are going to freely ask, nor what answer I'm going to freely give unless they actually sit us down and ask us--but by that point, they're actually in the room, and that changes everything! If they stay outside the room, they can't know what's happening inside because what's happening is our free choice. All they know is: at the end of the day, when we come out of the room, if they ask you: Hey, did Matt say UP or DOWN, or whatever, in answer to your question? And then they ask me: Hey Matt, did you say UP or DOWN in answer to the question? Then your answer and my answer to those questions will always agree 100% of the time--that's not free. And so it comes about that, according to quantum mechanics, it's thinking about freedom vs. determinism as an exclusive binary opposition that's the gigantic mistake: seen from the proper perspective, "determinism" is just the flip side of "freedom." What appears to us as determined is just the result of previous free decisions made by ourselves and others. To wit, quantum mechanics is the middle path through the millennial philosophical thicket. Okay, you may be wondering, but how do you get probabilities out of all this? Aren't probabilities what quantum mechanics is supposed to calculate? In the simple example given above, the electron would have to assign 33% X : 33% Y : 33% Z to your possible questions about its spin, while you'd have to assign 50% UP : 50% DOWN to the electron's answer to your question. Later, after leaving the room, you and the electron would each have to assign 33% X : 33% Y : 33% Z to my questions about what happened inside (remember, there are three different ways to ask about spin: X Y and Z), and I'd have to assign 50% UP : 50% DOWN to both of your answers. But I also have an extra degree of freedom: I can ask you X, and the electron X I can ask you X, and the electron Y I can ask you X, and the electron Z, etc. And so, I can consider assigning probabilities to whether or not your answer agrees or disagrees with the electron's answer in each of these cases. And in general, I'd have to assign even odds, 50% AGREE : 50% DISAGREE, to each case. BUT: if I knew your intention, that you were going into the room to ask the electron about its spin, then I could assign 100% AGREE : 0% DISAGREE to the cases where I ask you guys the same question afterward. Here's the whole table: XX: 100% AGREE : 0% DISAGREE XY: 100% AGREE : 0% DISAGREE XZ: 0% AGREE : 100% DISAGREE YX: 100% AGREE : 0% DISAGREE YY: 100% AGREE : 0% DISAGREE YZ: 0% AGREE : 100% DISAGREE ZX: 0% AGREE : 100% DISAGREE ZY: 0% AGREE : 100% DISAGREE ZZ: 100% AGREE : 0% DISAGREE (Just so you know, there are various subtleties I'm eliding over. Depending on what you decided to measure (X Y or Z), I might have to flip the table around--but perfect disagreement is as good as perfect agreement for our purposes!) Furthermore: in the general case, where I don't know your intention beforehand, I can't draw any conclusions about what happened inside the room from the answers that you and the electron give--whether you did actually ask the electron about its spin or not--because even if you hadn't asked the electron a question, but just sat there instead, and afterwards, I decided to ask you both the X question, and you agreed with the electron about the answer--well, it could just be a coincidence! Only if I get you guys to go back into the room, interact in the same way, come out again, and submit to my questioning a bunch of times, can I deduce what's happening inside the room, and be able to assign the proper probabilities. Now imagine some complicated situation of a bunch of guys interacting in a bunch of complicated ways, and so their answers will all have to agree in equally complicated ways depending on what they chose to ask each other and how they chose to answer. (And to point out the obvious: a guy can freely let answers to previous questions determine what questions they ask next, and so forth.) Afterwards, you sit all the guys down and ask them a bunch of your own questions and record the answers. Now assume that you can do this a bunch of times. By which I mean: assume you can get the guys to go back to where they started, and do it all over again. And when I say, "get the guys to go back to where they started," I mean that literally: you'd have to erase their memories, everything. So in physics, we usually just throw the old guys out, and make some new guys to replace them, which start from the same initial conditions, and then we run each set of guys through the same "apparatus" each time. Which apparatus and which starting conditions are our free choice. For example, the guys might be electrons and the apparatus might be a Stern-Gerlach device that applies an asymmetrical magnetic field in the X Y or Z directions (the "question") to the electron zipping through it, and then the electron smacks into one of two detectors at the end, one placed above the other, which clicks (the "answer"). And if the one above clicks, you learn: the electron decided its spin was UP! And if the one below clicks, you learn: the electron decided its spin was DOWN! So you make your various sets of guys go through some whole rigamarole a bunch of times, and because you decide what "apparatus" to send them through, and how the guys start off before they go through it, you restrict the kinds of interactions they can have inside, the kinds of questions they can ask each other and the kinds of answers they can give. But within this structure that you've determined for them, their individual questions and individual answers are free. Every time you run the experiment, all your guys make their free choices, and then afterwards, you sit them down and ask them some questions, and record their answers. And in the limit as you do this a bazillion times, you find that the answers, each individually free and unpredictable, always appear in certain proportions over time because you artificially restricted their freedom, narrowed down the possible ways they could get entangled, and you did the experiment enough times that your guys probably explored all the possibilities available to them within the limits you imposed. Finally, if you didn't know what apparatus you were sending your guys through, you could try to deduce that from the probabilities. To recapitulate: "interaction" or "measurement" or "asking a question and getting an answer" or "making a free decision and getting a free response from nature" is just "entanglement" seen from another perspective. To ask someone a question and to learn the answer is to become entangled with them. Nature ingeniously keeps track of the histories of how we've all interacted, what free answers we gave to what free questions, and makes sure that any new questions and answers in the future remain consistent with all the other relevant questions and answers in the past. And that gets pretty complicated to keep track of from a book-keeping perspective. But the essence of it, I think, is pretty intuitive. Or it would be if you didn't doubt, as perhaps you do, that you have free will. It should be clear: you aren't free from your circumstances, your limited capacities, or from the consequences of your decisions, but each decision you make is your free choice and uniquely yours. It's your private "subjective" choice in the sense that nobody in the universe can know anything about it until you decide it for yourself and then let us know. To me, this really does imply that we ain't robots, we ain't living in the matrix, that the world is best thought of as a community of free "observers" who get to explore a playground that they mutually and freely create for each other, whose one "rule" is that everyone gets to be equally free, each in their own place, which place is determined by the free choices of all: a cosmic democracy. In the best case, it's a playground; at worse, well, one thinks of that line of Jean-Paul Sartre's: "Hell is other people!" Now I'll be honest: I don't think the word "observer"--intentionally chosen by physicists to be scrupulously neutral with regard to its connotations--really does justice to the concept we're dealing with here. And I know using a word like "soul" is bound to make people uncomfortable because of its association with religion, metaphysics, superstition, whatever. But frankly, I want to make you uncomfortable. For almost a century now, our civilization has basically been in denial, blindly tottering on with its largely materialistic and deterministic view of reality, as if all the great experimental discoveries of quantum physics never happened. And that's a huge mistake, for us, and really, for the planet as a whole. Supposedly "no one understands quantum physics" or "anyone who tells you they understand quantum physics is lying" or "you can't understand anything about quantum physics without a PhD in math" or whatever. According to some people, quantum physics marks the moment that mankind discovered a theory of reality so crazy and divorced from our common experience that it simply couldn't be comprehended by our puny human minds, which are, after all, the result of a process of natural selection "optimizing" us for some environment, and why should we be optimized to understand electrons, photons, whatever, quantum things which are so absurdly tiny relative to us. Give up interpreting these mathematical symbols, and be content with the fact that you can use them to calculate with staggering accuracy the probabilities for stuff to happen if you set up some apparatus in a lab the right way! And then what? Accept fate? Free Will Theorems aside, there's a huge contradiction in this line of reasoning. Your refusal to interpret quantum physics doesn't change the fact that from day one we evolved in a quantum universe and not a classical one. And it's a frankly embarrassing fact that most accounts of natural selection, especially those targeted to popular audiences, still tacitly make the same assumptions about the world that Darwin did back in the 19th century. And sure, there's various theorems that show: when stuff gets large and complicated and interacts with other stuff in lots of ways that you can't exert any control over, and you lose track of who is entangled with whom, and lots of your stuff goes flying off into the void, the math says, in that case, from most perspectives, the probabilities you can calculate are indistinguishable from the probabilities that classical, deterministic physics would give you. And you might say, well, the human body is so big and hot and messy, it's gotta be classical in this way--and so--yay?--effectively, despite quantum physics, we're still robots, and the brain is just a classical computer, which deludes itself into thinking it's free because the fiction of a self proved useful in its classical evolutionary history. Then you invoke things like "Turing's Halting Problem" or "Gödel's Incompleteness Theorem," to show that: although sometimes you can use the results of a simple classical computation (A) to predict the results of a complicated classical computation (B), saving you the trouble of actually running B, in general, this isn't true--in general, there's no short cut to B save actually running B itself--in general, a simulation can't be faster than the real thing--so that therefore, even if we are a classical computation, in the general case, there's no way for anyone, including ourselves, to predict our behavior faster than us actually just behaving. And it's in this sense that we have "free will," now with scare quotes. (But then, of course, we have to knock our heads against the question of how the "observer" of quantum mechanics, our so-called "consciousness," and the dumb meat of our robot brains all relate to each other...) But fine. No doubt all these ideas have an important role to play for us. But let's go back a few steps. The shaky bit of reasoning is: "Well, the human body is so big and hot and messy, it's gotta be classical in this way." There's no reason to believe this. Evolution solves hard problems all the time, because it takes its time, that's kind of the point. And let's not forget that way back at the beginning of things, everything was tiny, and then we scaled up to our present size. And in the course of history, if some "quantum trickery" allowed one little guy to outcompete some other little guy, wouldn't the little guy who could pull the quantum trickery win out in the end? And as the little guys get larger and larger and more and more complicated, wouldn't the guys who could still pull some quantum trickery, despite the fact that they might be large and complicated, outcompete those who couldn't? So that, actually, natural selection would predict that we've evolved to exploit as much quantum trickery as possible, given our history. More bluntly: we live in a free universe, and so it's quite likely that if we're optimized for anything, we're optimized for dealing with that freedom in interesting and useful ways. And maybe this explains why most people have to be educated out of their "naive" belief in their own free will. And it also probably explains why, as is becoming increasingly clear, the mathematics which describes human communication (language) has striking similarities to the mathematics of quantum mechanics, as most of our interactions are mediated through language, after all. And one imagines, in the future, describing human society in terms of its entanglement structures, and so on. (Sidebar: As a writer, I can't hold back from pointing out that much of 20th century literary theory has to be entirely rewritten. It's not fictions all the way down. Whereas, reading standard literary theory, you might think you could never trace back a "text" to a particular particular "author" with 100% certainty--well, if you thought that, you'd be wrong. Sometimes you can! After all, communication is just entanglement from another perspective. No text is an island, a bare sequence of letters or 0's and 1's. Rather every text is just the tip of an iceberg floating in a sea of relationships that remember the story of our mutual freedom. Metaphors aside, this fact is actually what makes quantum cryptography, perfect unbreakable encryption, possible. There are some things in the world that cannot be faked, cannot be forged, because otherwise our freedom wouldn't be preserved. Not everything is fake news. And this shit is happening today: http://www.catalannews.com/highlights/item/catalonia-aims-to-lead-quantum-revolution ). Yet despite the backbreaking labors of a great many scientists and philosophers over the last century, our civilization still seems largely to regard the "free will" vs "determinism" debate as open, with the safe bet being determinism, especially among the highly educated (or the highly religious who believe in "predestination," and so forth). Sure, the math is hard, but I don't think that alone can explain such a mass delusion. As I say, quantum theory is actually breathtakingly simple and intuitive, but only if you approach it as a theory of freedom. In fact, I'd even drop the word "quantum," and just call it the theory of freedom, the theory describing how our free decisions determine the circumstances in which yet more free decisions can take place--and all the spookiness and weirdness just comes from the fact that nature bends over backward to maintain our freedom for us. So why is it that the theory of freedom is so widely perceived as strange, esoteric, spooky, and weird--like we're all trying to collectively repress it or something? Well, historically speaking, there are aspects of modern science that are reminiscent of "magic," at least as far as the early pioneers of the Scientific Revolution--or the theologians of the Protestant Reformation-- would have been concerned. Also, in the modern period, it seems that the rejection of free will became a mark of prestige and sophistication among the highly educated elites of Europe and America, a way of distinguishing themselves from the common run of people. Furthermore: just as among a certain class of religious fundamentalists, it became common sense that everything was, is, and always shall be absolutely predetermined in every particular by God's divine plan, it became common sense among a certain class of anti-clerical intellectuals that any talk of freedom, in a metaphysical sense, is just a way for powerful people to fuck over the little guy, distracting them with metaphysical fantasies from becoming aware of their outward oppression and unfreedom, in the "real" world, the material world. (I mean, they had a point.) Turning to the purely psychological, a lot of people go into science because of a desire for certainty and control. In that sense, at least, it's not that different from religion. It's nice to think the world is governed by rules, and if you follow the rules, everything will be fine. Whereas if it turns out that, in some sense, we get to change the rules... we suddenly have to start thinking for ourselves, about what we really want, and that's a lot to deal with! If it turns out that our circumstances are determined not by eternal law per se, but rather by the free decisions of other free beings like ourselves, that means that we are never truly isolated or independent from those with whom we share this existence. We're all in it together, for better or for worse. Finally, and more ominously, it's certainly easier to sell shit to people and convince them to sacrifice their lives on the altar of money, if they're convinced they live in a "material" world, and that's just the way the cookie crumbles. I mean, precisely because we are free, we can give up our freedom, and allow others to determine our decisions for us. And, well, our civilization has a vested interest in allowing slavery to continue because it is very profitable! Okay, enough moralizing! Earlier I suggested some changes in our terminology. Quantum theory ought to be termed the theory of freedom, free theory, or freedom theory, however you like; and the observer ought to be termed the soul. But what is a soul? Beats me! It is immortal? How the heck should I know?! Whatever it is, though, physics says you're one of them. Electrons are too--it's not a specifically human thing. I vote we bite the bullet and just realize that soul doesn't have to be a religious concept, it can be a scientific concept too, and it can be used in an entirely agnostic, even atheistical way, if that's how you roll. And the concept signifies: one's irreducible uniqueness, one's private domain, and one's essential freedom. And we can leave it for tomorrow to figure out what it all means. But let's be honest about what spooks us, and deal with it together, out in the open. The philosophy of yesterday was all about dealing with the fact that we thought we lived in a world of deterministic laws--what we should do about that, how we should live our lives in the light of that. And it told mankind a story about how some robots evolved this convenient fiction that, in fact, they weren't robots, but free souls. The philosophy of tomorrow will be all about dealing, instead, with the fact of our freedom and our entanglements--what we can do with them, which, I'm sure, we haven't even begun to imagine. It will have to tell mankind a story about how some free souls evolved this convenient fiction that, in fact, they weren't free souls, but slaves.
- SECOND MEETING
Hey all! I hope everybody is well, and to those who came to the first meeting back on the 29th, thanks again, and I hope you've had the chance to mull over everything we talked about! I had a lot of fun, and I'm looking forward to seeing all of you again. I apologize if the meeting was a little scattered--but I believe it was a necessary evil, just so I could rapidly get a sense of where everybody stands in terms of physics background, math background, philosophy background, history background, etc. In any case, hearing the kinds of questions you asked, and how you responded to my answers, was incredibly helpful to me, and I've spent the last few weeks trying to organize all my material to best help y'all, so that we have a definite plan when we regroup, and we can get on the same page as quickly as possible. So for our second meeting, I'll lay out very clearly: a) what are the mathematical prerequisites for understanding modern physics, and why b) what are the physical experiments that supposedly confirm the theories c) what are the issues with interpreting the results, in as succinct a way as possible. I'm flexible with the timing of this meeting, so if anyone has any objections, please raise them. Just to whet your appetite, let me give you a quick outline: The first thing you have to understand is that the guiding spirit of modern science, that shapes its style of argument and its standard of proof, is symmetry. This was always an undercurrent in the history of science, but it’s been explicit in physics since the days of Einstein. Physics is the study of the symmetries of nature. It's from identifying symmetries that we can make predictions, do calculations, develop a "picture" of the world. And all of modern physics, the math, the experiments, whatever--it can all be derived from the consideration of actually just 7 symmetries! So what are they? When you close your eyes, you don’t know where you are. When you close your eyes, you don’t know what direction you’re facing. When you close your eyes, you don’t know if you’re at rest or moving at a constant velocity. When you close your eyes, you don’t know what time it is. When you close your eyes, you don’t know how someone else has described the world with a set of symbols: you only know your symbols. When you close your eyes, you don’t know if you are matter or anti-matter, moving forward in time or backwards in time, with a positive charge or a negative charge, the source of a field or a sink. When you close your eyes, you don’t know if you are at rest or accelerating. If you feel tidal forces (relative accelerations) across your body, you don’t know if they are the result of your interactions with things, or the result of the curvature of spacetime (the relationships between things). And that’s it. From those 7 observations about the symmetries of your inner subjective experience, you can derive all of modern physics. Consider that each of the 7 symmetries can be rephrased in terms of the relativity of some quantity, such that the quantity can only be described in terms of two or more things. Distance is relative. Orientation is relative. Velocity is relative. Time is relative. Representation is relative. Being a “plug” or a “socket” is relative. Acceleration is relative. We can also rephrase the 7 symmetries in terms of some quantity which doesn’t change in time (in other words, it relates two or more times). Conservation of Identity. Conservation of Angular Momentum. Conservation of Linear Momentum. Conservation of Energy. Conservation of Probability. Conservation of CPT. Conservation of 0. [What's CPT? For example, C (charge) could be +1 or -1 (positive or negative). P (parity) could be +1 or -1 (the thing or its mirror reflection). T (time) could be +1 or -1 (forward or reverse). You multiply C*P*T and that’s what's conserved.] Consider what a symmetry means. Symmetries relate perspective switches to actions. The simplest example is distance. I move forward, I see you get bigger / you see me get bigger. I move back, I see you you get smaller / you see me get smaller. You move forward, you see me get bigger / I see you get bigger. You move back, you see me get smaller / I see you get smaller. Therefore anything you can do, I can undo. If I move forward and you move back, we can fix a distance between us, so that we stop changing in size. Similarly, if I walk in a circle around you in the center, we can always rotate to stay face to face: we can fix a face between us. And in general, we can use each symmetry to fix a “coordinate” that relates me and you: when we are face to face, we call that 0. This is where coordinates come from. So, I guess at this point we have to talk about math, which is the rules for working with coordinates. Each kind of coordinate has a different kind of mathematics, and so there are 7 kinds of numbers that we have to use to write down our theories of physics: 0/1, Counting Numbers, Integers, Rationals, Complex Numbers, Vector Spaces and Linear Transformations, Fock Spaces and Ladder Operators. Each kind of number can in some sense be derived from those which come before it. Creation "iterated" leads to counting up/counting down; counting up/counting down iterated leads to addition/subtraction; addition/subtraction iterated leads to multiplication/division; multiplication/division iterated leads to exponentiation/root-taking; exponentiation/root-taking iterated leads to tensor product/inner product; tensor product/inner product iterated leads to the "direct sum of tensor products"/"inner product of direct sum of tensor products," and that turns out to be enough, as far as we know, to describe the mathematics of symmetry used in physics. The reason, essentially, has to do with the fact that "symmetry groups" can be represented as "linear operators on vector spaces": this branch of mathematics is known as "representation theory." All this mathematics was discovered for completely objective reasons by mathematicians in the 19th century. And then the physicists of the 20th century used that mathematics to experimentally prove that: not only do we have these 7 symmetries of our internal subjective experience, not only do we have 7 kinds of numbers to describe those symmetries objectively with mathematics, but also that *the laws of physics themselves are symmetrical under those 7 symmetries*. It was proved by experiment between[masked] that: the laws of physics don't change if you do any of the following things: Continuous spatial translation; continuous spatial rotation; continuous temporal rotation (which changes velocity and clock rate); continuous temporal translation; continuous "phasing" of internal degrees of freedom; discrete permutations of the symbols we use to denote coordinates; discrete charge conjugation/mirror flip/time reversal; and finally, any continuous transformation that depends on spacetime coordinates. Symmetries of spatial translation and spatial rotation were known to people in antiquity as the laws of perspective drawing. Galileo is credited with discovering the the symmetry of temporal rotation, and Einstein for clarifying temporal translation with special relativity. You might say Hermann Weyl should get credit for symmetry under continuous phasing (a kind of rotation) of internal (private) degrees of freedom, since he came up with the idea of a “gauge theory,” but really it was all early pioneers of quantum mechanics and the ferment of the time—Bohr, Born, Heisenberg, Schrodinger, Pauli, etc—who nailed down 5 and 6. 5 specifically takes you to consider quantum field theories: fermions (like electrons!), bosons (like photons!) Bosons, for example, are completely symmetrical under permutations of the labels aka the symbols used to denote them: this allows them to bunch together all into one state: a laser beam, for example! Fermions, however, end up facing the wrong direction when you permute them, and by exceedingly clever argument, this proves that 2 electrons can’t share exactly same state, and therefore they have to stratify into electron orbitals. From this literally all of chemistry can be deduced. The electrons determine the identity of the atom, and how it can bind to or “entangle” with other atoms: the photons mediate the inter-atomic interactions. It’s worth noting that the theory for the experiment which would prove that two spacelike separated events are not necessarily statistically independent, in other words, that quantum entanglement exists, was devised in the 1960’s by John Bell, and various iterations of the experiment were carried out in the 1970’s, 1980’s, up to the present, at which point all loopholes are considered closed. Going back a little bit, the demand that quantum field theory be consistent with special relativity led to Dirac’s discovery of anti-matter. CPT symmetry was proved in the 50’s, and indeed by then Feynman and his whole generation had begun the big experimental project of the 2nd half of the 20th century: particle physics. The final symmetry, any continuous transformations that depends on spacetime coordinates, has a special status. It was stated by Einstein: the principle of general covariance, the equivalence principle, the identification of gravitational and inertial mass, and is the basis of general relativity. Yet general relativity is essential a theory of public reality. It doesn’t incorporate naturally into its framework the private internal degrees of freedom introduced in 4, 6, and 6. So despite the fact that 7 was discovered, in a way, before quantum mechanics existed, nevertheless it’s the one we’re still working out (though the shape of it is clear). And we should give Isaac Newton credit for gravity too, since gravity is precisely the result of the laws of physics being symmetrical under any continuous transformations that depends on spacetime coordinates. Why? Well, spacetime coordinates are “fictions,” so to speak, “symbols,” developed by observers to coordinate their experience. If we all agree to make decisions based solely on what can be coordinated by those symbols, then we’ll remain at rest (or “inertial,” more generally) relative to each other—we’ll remain face to face, we’ll remain coordinated, despite the fact that we might individually undergo changes. If, however, we begin to make decisions based on the specificity of our symbols—on the accidents of our representations—then we will lose our overall coordination—and, as it were, begin to drift, either away from each other or towards each other, relative to our spacetime coordinates. When this happens, we say we’re gravitating. So that's modern physics in a nutshell! It's all about symmetries, and the way the free acts of nature are conditioned by them. Modern physics dispenses with the idea of determinism. It dispenses with the idea that the world is like a computer program being inevitably unpacked from a sequence of 0's and 1's, and that the goal of physics is to discover the ultimate computer program that generates the world. The laws of physics are not instructions for how to build a world from first principles: rather, the laws of physics describe the symmetries by which we can coordinate our experience. Hope to see all of you soon! All best, Matt