thoughts on gymnastics and Hinduism

[spoonless]

Gymnastics has started up again, I am super excited about that. They got rid of Saturdays for the summer, and I'd been waiting all summer for the fall to come so I could start going again. Went the first time on Saturday, and... oh so good. It's funny, I've been thinking a lot lately about what I want out of life, and have been realizing how totally lost I am. But when I sit down and try to figure it out, I always come back to the only thing that I know for absolute sure I want... which is to be able to do gymnastics every day of the week. I can't accomplish that in the town I'm living in--the YMCA is so sick of me bugging them about it, they no longer even return my phone-calls. And the other gymnastics programs in town are similarly kids only. I don't mind driving up to Chicago once per week for the time being, but I will eventually need more than once a week. I'm ok with waiting though. But as a long term goal, I do hope to move toward that within the next 10 years.

I spent 10 hours on Sunday reading Wikipedia articles on Hinduism. I've had this happen to me before, reading too much Wikipedia in one day--I've noticed it's one of the few things lately that I'm somewhat dangerously addicted to. I broke for 30 minutes to eat lunch, although for some of that half hour I just brought my food over to the computer to read while I was eating! Literally, when I get interested in a topic on Wikipedia, I find it's almost impossible to pull myself away from the computer, I just keep saying "1 more click, I have to read this link and then I'll go do something else!" Fortunately, as opposed to most weekends where I've got lots of stuff I wanted to get done, I really did have enough time to relax that weekend, and reading Wikipedia is one of the most relaxing things for me. Normally, 10 hours of Wikipedia reading I would consider "way too much" but it was only slightly too much this time because I got a lot out of it and enjoyed it.

The more I read about Hinduism, the more I realized how much more complex it is than any other religion I've encountered. Just trying to find all of the names of the various sacred texts involved took me through at least 10 pages alone, and I'm not quite sure I've found them all (and certainly don't remember them all). The main ones that stick out in my mind though are the 4 Vedas, the Upanishads (there's about 200 of them!), the Brahma Sutras, and the Puranas. This only scratches the surface though, and as I say, I don't even remember most of the names of the texts that I read about, and I didn't even read any of them directly (although I did briefly open up one of the Upanishads to see what it looked like). I also don't think I even scratched the surface on getting a handle on all the different variants of Hinduism. Unlike much simpler more narrow coherent religions like Christianity, there is so much more variation within it. For example, there are monotheistic sects of Hinduism, there are polytheistic sects, and there are atheist and/or pantheist sects. There are dualist versions of Hinduism, and also non-dualist versions.

So far, the one that stands out as the most resonant with me and interesting to me, is Advaita Vedanta. (Although that's not to say there isn't some other version that would be better that I haven't found yet.) This interpretation actually makes so many of my confusing thoughts about how to view deities make more sense. In Advaita Vedanta, Brahman is absolute, ultimate reality... the continuous whole that has no attributes, boundaries, form or shape of its own. In physics, you'd call this the "unified field theory" or the "theory of everything". Pantheists would call this "God" but not regular monotheists since it is not a someone but a something. The problem is, Brahman is an abstraction that stands for everything but as such has no properties or attributes to distinguish it from anything else... since there *is* nothing else. So how can you possibly comprehend it? There's not really any way to actively meditate on it or to worship it. The solution, according to Advaita Vedanta, is for humans to personify Brahman by projecting animate qualities onto it, which then becomes an omnipotent being called Ishvara (essentially, the same as God or Allah from monotheistic religions). But this still leaves Ishvara mostly formless and personalityless. So you have to project more qualities onto Ishvara, which then splits Him into the holy trinity (or as Hindus call it, the Trimurti)... Brahma, Vishnu, and Shiva (creator, maintainer, and destroyer... in physics, these are called "creation operator", "propagator", and "annihilation operator"). These three deities then each have 11 eminations, 11 different even more concrete ways in which they can appear to humans on earth. (On a side note, it's a somewhat interesting numerological coincidence that string theory (well, really M-Theory) requires 11 dimensions of spacetime, and the Qabalastic tree of life involves 11 Sephirot; I'm not sure if anyone has tried to line either of these up with the 11 forms of each Hindu god though). Similarly, there are 3 goddesses, the Tridevi: Lakshmi, Parvati, and Saraswati. The female counterparts of the holy trinity. They also each have 11 eminations. This leads to a total of 33 gods and 33 godesses, although it's interesting that this interpretation sort of combines atheism, pantheism, theism, and polytheism all into one unified theory. It makes a lot of sense to me posed in that way. The abstract formless attributeless Brahman is more true but less useful than if you split it into more concrete manifestations and worship those individually. (Note: some of the specifics on how the gods and goddesses divide up I may be confusing with other versions of Hinduism, but putting together lots of different stuff I've read, some a while back, this is the most coherent picture I can come up with.)

Another thing I really like about Advaita Vedanta is that it's explicitly non-dualist. There are other versions of Hinduism (for instance, Dvaita Vedanta) where they succumb to dualism, and believe--as Descartes mistakenly did--that there is both a spirit world and a material world, or at least both of those different kinds of substances in the world. This is something I think modern science has pretty thoroughly refuted. I knew that Buddhism for the most part was non-dualist, but it's encouraging to see there are threads of Hinduism that are as well. Unfortunately I get the sense that most people who believe in Advaita Vedanta are more like idealists than materialists, but in some ways that seems like a minor difference compared to the difference between dualism and monism. (After all, in some ways you could just view it as different take on what the best word is to call the single substance that makes up everything... and admittedly there are good reasons why really neither of the two standard choices offered are quite right. I just happen to think that materialism is the clear cut answer if you had to choose one or the other).

Also, I really like the phrase "Ayamātmā brahmā", something from the Mandukya Upanishad. It means "This Atman is Brahman". Atman is the soul of an individual, and Brahman is the whole universe (or God, in the pantheistic sense). It identifies part with whole in a very holographic way. I haven't had this thought about any phrase since I got my first tattoo, but suddenly I had the thought that this would make a really awesome tattoo! Not that I would get it without doing more research into what it really means and how it connects to other things. But it's a neat idea, and very parallel to similar thoughts I've had over the past few years. Strange that all of this was just sitting here all of the time, and I never read it until now... putting these different pieces together.

Comments

[easwaran.livejournal.com]

I was just talking with Matt several times in the last few weeks about the Schrödinger equation - I had never calculated solutions for it before, and he didn't know what it meant that the set of possible wave functions was an infinite-dimensional vector space.

Anyway, I thought it was interesting that even for the toy models (the completely free particle, the particle in a box, the particle in a finite potential well), calculating the solutions requires some awful hand-waving. At least, in the discussions in wikipedia, they claimed that there are sinusoidal solutions to the equation for the completely free particle, even though the sinusoids are just as non-normalized as the exponentials. And for the other two, we have to assume that the space of possible wave-functions includes ones that have discontinuities in certain derivatives at certain points, in a way that intuitively makes sense, but I saw no mathematical justification for.

[spoonless.livejournal.com]

Yes, it's ironic that for the two most commonly used operators used in quantum mechanics--the position operator and the momentum operator--neither of them has eigenstates (for the free-particle case) that are actually in the Hilbert space. The momentum eigenstates are sinusoids, and as you say, non-normalizable (and therefore not physically allowed). While the position eigenstates are Dirac delta distributions, which are not even functions! (They're just the limit of a series of functions.) So neither of them are actually in the space you're supposedly working in. And neither of them ever happen in the real world.

There's a fairly intuitive way to understand why they never happen in the real world though. You can't localize a particle to be at an exact location, because then the uncertainty in the position would be zero, which would yield an infinite uncertainty in momentum. Conversely, you can't have a perfect sinusoid that stretches out all the way to the boundary of spacetime (infinite uncertainty in position) which is what would be required in order to perfectly measure the position of something.

The moral of the story is just that you can never precisely measure position or momentum, you always have to measure some combination. The position and momentum operators are mathematical idealizations of real measurements that you can do which consist of combinations of them.

[easwaran.livejournal.com]

Why can't you have a perfect sinusoid that stretches out across all of spacetime? (Note - if we're treating spacetime as R^n then it has no boundary, so it's not stretching out "all the way to the boundary of spacetime".) It seems like your intuitive way of understanding this presupposes some sort of uncertainty principle, but I thought this was supposed to be proving the uncertainty principle rather than presupposing it.

Also, I guess I hadn't realized that this was supposed to have some connection with measurement - I thought the Schrödinger equation was just some equation that the wavefunction should always satisfy (though it's unclear to me why there should be some constant that we can call the "energy" for which the wavefunction is an eigenvector).

[spoonless.livejournal.com]

Why can't you have a perfect sinusoid that stretches out across all of spacetime? (Note - if we're treating spacetime as R^n then it has no boundary, so it's not stretching out "all the way to the boundary of spacetime".)

An actual wavefunction "in the real world" is supposed to be an indicator of the probability of finding the particle at each location if you were to measure its position. So physical wavefunctions have to be normalized to 1, otherwise the whole interpretation in terms of probability would fall apart. This means that if you're working in a space like R^n it has to fall off to zero as you approach infinity. You might be able to find a mathematical function that obeys the Schrodinger equation, but if so it cannot correspond to an actual state the particle could be in. It's sometimes called a "non-physical" wavefunction because it fails to satisfy one or more requirements that real wavefunctions satisfy (like being normalized to 1).

Incidentally, I think if you are working in a space that has a finite volume then this is not an issue. By "boundary" I just meant "out to infinity"--I see why this is a very confusing word to use here and I should have avoided it. I think the reason why I (and others) are used to using "boundary of spacetime" to mean infinity is that often the only way to formulate things in QFT is to work in a finite box and then take the limit as the size goes to infinity... although you still make sure the values of all fields (and their wavefunctions) go to zero at the boundary, otherwise you'd end up with a big problem when you tried to make the size infinite. Only if you were working in such a box (or some kind of compact space) and not intending to take any limits would you be allowed to have a non-zero wavefunction at the boundary. Of course, this *could* be the situation in real life, so if that's the case then pure sinusoidal functions could be perfectly physical--however, in practice you still wouldn't really expect to ever find one that stretches all the way across the universe. Particles travel in wavepackets, they are typically localized to pretty small regions.

[easwaran.livejournal.com]

Particles travel in wavepackets, they are typically localized to pretty small regions.

This doesn't make sense to me - I didn't think anything other than a perfect sinusoid or exponential was a solution to the equation, so I don't understand how these wavepackets work. Is it only because actual particles are always in the presence of a potential field of some sort?

[spoonless.livejournal.com]

See my other comment regarding energy eigenstates (stationary states). Most physical states are superpositions of stationary states, not stationary states themselves.

I suspect the version of the Schrodinger Equation you're talking about is the time-independent Schrodinger equation. All wavefunctions obey the time-dependent Schrodinger equation, but only stationary states obey the time-independent. The time independent Schrodinger equation is really just a convenient way to find the stationary states of a system. Stationary states have trivial time-dependence (all that happens is their phase rotates in time, while their magnitude remains constant). These are then used to build up the rest of the states of the system, most of which have non-trivial time dependence. A typical state in the real world would be a wavepacket, where the size of the packet spreads over time as it travels along.

[spoonless.livejournal.com]

Also, I guess I hadn't realized that this was supposed to have some connection with measurement - I thought the Schrödinger equation was just some equation that the wavefunction should always satisfy (though it's unclear to me why there should be some constant that we can call the "energy" for which the wavefunction is an eigenvector).

It's tough to find anything in quantum mechanics that isn't related to measurement somehow =) You don't have to think about measurement here, I guess, if you don't want to. It's just that the reason position and momentum are usually chosen as nice bases to work in, because they correspond *roughly* (but not exactly) to common types of measurements that are done. The Schrodinger equation in its abstract form is an operator equation, but the more commonly used "wave equation" that is also sometimes referred to as the Schrodinger equation (and was what Schrodinger was initially working with) is "the Schrodinger equation in the position basis". You can write it down in any basis, or you can just write it as an operator equation that is basis-independent. Each basis has a different set of eigenstates associated with it, that usually correspond to physical observables.

Regarding energy, in one sense it's no different from any other physical observable. The only thing that makes it special is that it is the conserved quantity associated with time-translation. Any system which has laws of physics that have no explicit reference to a specific time (like say, the big bang) must conserve energy, because of this relationship. This is true both in classical mechanics and in quantum mechanics, but one implication that has in quantum mechanics is that eigenstates of the energy operator (Hamiltonian) are time-independent. That's why energy eigenstates are special, they are sometimes referred to as "stationary" states since they do not change over time like most states do (actually, their phase can change, just not the magnitude which determines the underlying probability distribution).

Incidentally, when you say "for which the wavefunction is an eigenvector", that's not true of most wavefunctions. Most wavefunctions are not an eigenstate of energy... only stationary states are. However, stationary states are usually the first thing you want to find when you're studying a new quantum system, since it's very easy to build any more complicated time-dependent state out of a superposition of the simpler stationary states.

[easwaran.livejournal.com]

I'm skeptical about the amount that measurement ought to be central to the theory - scientists seem to be attracted to this view for bad metaphysical reasons. But I don't really know.

Most wavefunctions are not an eigenstate of energy... only stationary states are. However, stationary states are usually the first thing you want to find when you're studying a new quantum system, since it's very easy to build any more complicated time-dependent state out of a superposition of the simpler stationary states.

Ah, that's useful to know. So there are in fact plenty of states that don't have a well-defined energy in this sense (where the energy is the eigenvalue). That's useful to know! This didn't seem to be mentioned in the wikipedia article about the Schrodinger equation.

[spoonless.livejournal.com]

Anyway, regarding the position and momentum eigenstates, this is a typical pattern that happens... you pick some case where the math is easy to work out, even if it's a huge approximation and not realistic. Then use that as a conceptual tool to understand more complicated cases.

In this case, I'm not sure I would even call it handwaving, because it has been worked out pretty rigorously... it's just that the details of it are never taught to undergrads, or even grad students usually.

[easwaran.livejournal.com]

But I guess it trains physicists to accept handwaving in the mathematics.

[spoonless.livejournal.com]

I think this is true--it gets students used to the fact that physics is complicated enough that you will never be able to understand it if you try to derive every detail in a step by step proof. You have to learn to skip steps sometimes and reason by analogy, and trust that the work others have done in working out the details is sound. One of the hardest parts of physics is trying to know when you should be suspicious that some step was done in too cavalier a way, and when you should not bother verifying every step. You just have to develop a knack for spotting which kinds of handwaving holds up and which kind falls apart under scrutiny.