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Life in Infinite Dimensions

Hilbert spaces are weird.
Image: stuartpilbrow/Flickr

What could be more intuitive than the notion of space itself? Space is how we know here from there and all that entails. Which is everything. Imagine some existence without here and there and everything, all of us, shared a single-dimensional point, somehow. You, me, President Barack Obama, Taylor Swift—a single point in space.

It's just as hard to imagine the opposite, a space of more than three dimensions. A fourth dimension for time isn't so bad, where we might still imagine our three-dimensional world, but only with a sort of inner motion or current: moving without moving. Or maybe imagine time like a computer screen, a static reality that is always refreshing as time progresses. The frequency with which it refreshes, that's time.

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But five dimensions? Fuck.

There's an easier way to look at it, courtesy of my old Calculus III professor. We think of space as space, heres and theres, but these three dimensions are really a commingling of three variables, each one describing something different: length, width, height, x, y, z. There could be other spaces too, maybe one of red, green, and blue, or one of temperature, pressure, and humidity, etc.

So: three things connected by some equation of three variables. RGB, red-green-blue, color is properly considered as a space, the RGB color space. There are other color spaces, where we might imagine a color as a point given by three axes, with each one representing one constituent color. Usually, we imagine the color, not the space.

It doesn't matter what the variables, or axes, of a space are so long as they're related by some function, an equation relating all those variables. It's easier to see the possibility of other dimensions then: temperature, pressure, humidity, and then, say, water vapor pressure, mixing ratio, CO2 ppm, x, y, z, a, b, c. This doesn't make the graphical representation of extra dimensions easier to visualize, but the idea is maybe easier to get a hold of when it's not literal heres and theres.

(To really push the bizarre intuition of spaces, consider the additional feature that, as we add axes/variables to our function, each new axis has to be at right angles to all of the other axes. So, look at a three-dimensional graph and try to figure that out.)

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In the quantum world, things get really, truly wrong. Here we have spaces of infinite dimensions, by necessity. Spaces of possibilities.

First, imagine that a point in some space is described by a list of variables, and this list is stored inside of a vector, which is a just a stack of values, one for every dimension. To describe me in some 10-dimensional space, I could be described by the coordinates (x1,x2,x3 … x10) all lined up neatly in a vector, like a silo for numbers.

In non-infinite space, we can do things like find angles between these vectors and the vectors that might result if we add them together. We can do this for any finite number of dimensions, like a billion. There is an angle between two vectors with a billion dimensions.

Infinite dimensions doesn't mean quite what it feels like it should mean, which is some infinitely huge thing, like a train that doesn't have a beginning or end, just length that expands forever and ever. Infinity can be contained within something finite too.

When I play a note on my viola, that note can be described by a wave with some frequency and amplitude. So just think of a wave spanning space from point A to point B. The thing about waves is that they're always the sum of other waves. My note is a wave, but it can be described as two (or more) other waves, but also two other other waves where one wave is a little bigger and the other a little smaller. And so on.

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This decomposition becomes arbitrary at a certain point, which is a bit like saying it becomes infinite, or a space of infinite possibilities.

This also happens to be the mathematical description of Schrödinger's cat. It's not that the cat is both alive and dead, but that until we look at the cat, there's some chance that it's in either state. A probability: a bunch of possibilities together as one. This is what it is to know something indefinite (infinite), but real. A wavefunction.

So, when we have two things, particles or cats or whatever, and they're described by infinite dimensions (a wave vibrating in space), they can relate in such a way that together they're not so infinite. They converge to a real number, which can be found by taking what's known as an inner product. Take every dimension packed into each of those vectors, multiply the corresponding dimensions from each, and add it all up.

But, wait. How can you add up two vectors with an infinity of values? It only works if the two vectors' inner product converges. It helps to imagine our infinite vectors not so much as infinitely long lists of values (x1,x2,x3,x4, ...) but as functions or equations that give those values (f(x)). That might make our conception of "infinite dimensions" a bit more reasonable; they're infinite because they're not just lists of constant values, but boxes in which we can put in one number, any number, and get another in return.

If you take some function and keep plugging values into it, adding every new value to the previous total, sometimes you hit a limit, which is just a number. Infinity can in a sense just wind up being something stupid, like 3. The function never actually equals 3, but is still defined by it. So, you can take that inner product of two vectors and they might wind up being just a normal number. In cat terms, a normal number is "alive" or "dead." Maybe that takes some of the mysticism out of the whole quantum business.

Good.

Is all of this dimensional business more or weird than when we started? Infinity can be finite; space can be color; cats can be waves; math is still hard.