On Philosophy

April 9, 2007

Quantum Properties

Filed under: Metaphysics — Peter @ 12:00 am

The quantum world may seem completely incomprehensible under a common sense understanding of what a property is. Under such an understanding quantum physics seems to imply that objects only have properties when they are measured, and that at other times that they have only a weird superposition of properties, in which they both possess and lack any given property. But this is an artifact of certain unexamined intuitions about properties, or perhaps a problematic understanding of quantum physics. Under a better understanding of what a property is (as simply a certain causal disposition) and under a better understanding of quantum physics much of this apparent strangeness vanishes. Which is not to say that quantum physics isn’t a revolutionary way of looking at the world, it is, but it doesn’t have to be as counter-intuitive as it is sometimes presented as being.

Let’s start with a better understanding of how quantum mechanics works. Obviously the best understanding comes from working through the math, but since explaining the math would take a tediously long time, and require an understanding of linear algebra, I will stay away from it as best I can. If you are really interested I would recommend “Quantum Mechanics and Experience” by David Z Albert. A good place to start is by considering how quantum physics treats a measurable. To say that something is a measurable of a particle, in this context, simply means that we have some device or observer that responds in a number of different ways to the particle. For example, it might be a box with a needle on the side that swings one way or the other when a particle is passed through it. And furthermore we will say that something like this is a measurable only if the result is repeatable, meaning that if we put that particle through our box again, without disturbing it, the meter yields the same result. Normally we think of such measurables as properties, but that would actually be a mistake in the context of quantum physics, as I will show. Instead of considering a complex measurable like position, let us start with a binary measurable, one that can be only be found in one state or another. An example of such a measurable is the x-spin of an electron, which can be in one of two states up (↑x) or down (↓x). In quantum physics such properties are treated mathematically in terms of a two-dimensional vector space (a plane in which every point is labeled by an ordered pair of numbers). An electron’s being ↑x is represented by saying that its state is the vector ⟨0, 1⟩ (an line straight up from the origin) and being ↓x by saying that its state is the vector ⟨1, 0⟩ (a horizontal line to the right of the origin). Now let’s consider another measurable of the electron, its z-spin, which again can be up (↑z) or down (↓z). But the z-spin is not a completely different property from x-spin. It turns out that it is impossible for an electron to “have” both a z-spin and an x-spin, which is a way of saying that if an electron is measured to have ↑x then when its z-spin is measured there is a 50% chance that it will be measured as ↑z and a 50% chance of being measured as ↓z. And once its z-spin has been measured its x-spin is then indeterminate; we simply can’t have electrons that can be consistently measured to have both one particular x-spin and one particular z-spin. Mathematically, because of this and some other unusual quantum effects, the z-spin is also represented by the electron having a particular vector as its state. In this case ↑z is the vector ⟨1/√2, 1/√2⟩ and ↓z is the vector ⟨1/√2, -1/√2⟩. Because the state of the electron is represented by only one vector it is obvious from the mathematics that it can’t have both a particular x-spin and a particular z-spin. What is particularly interesting is what happens when we attempt to measure an electron whose state is not one of the vectors that the measurable is not defined in terms of. In quantum physics trying to take such a measurement “collapses” the state of the electron into one of the vectors that the measurement is defined in terms of. Which one is a matter of chance, the probability of which is the dot product of the electron’s current state vector and the particular vector that represents the electron being in that measurable state squared. In our example with x-spin and z-spin the mathematics is such that if the electron has a particular z-spin, say ↑z, ⟨1/√2, 1/√2⟩ then the probability of it collapsing into ↑x, ⟨0, 1⟩, is .5 and the probability of it collapsing into ↓x, ⟨1, 0⟩, is .5, which is how things turn out experimentally.

Hopefully you are with me so far, because there are only two more details that I need to add to the above description of quantum mechanics. The first is that an electron’s state vector can be any vector of length 1, not just one of the vectors that we have defined as a particular x-spin or z-spin. In fact many interactions with other particles adjust its vector only slightly. The second interesting twist is that x-spin and z-spin are not the only measurables for this vector space. It turns out that there are as many measurables as there are pairs of vectors at right angles, in other words, infinitely many. It is just that the easiest measuring devices for this vector space to construct happen to be for x-spin and z-spin.

Now with that out of the way we can talk about what the properties of the electron are. But first we need to say what a property is. Going back to my previous definition I will define a property as a particular causal disposition. Which in the case of quantum physics means that the state of the electron (its vector) is a property, some particular set of measurables. Each state vector has its own particular causal dispositions, specifically it has a unique probability of collapsing into each particular outcome for every set of measurables. And I think that this is also obvious from the fact that we can in theory construct an infinite number of measurables, one for each vector state. So where does the confusion come from? Well the problem is really one of terminology. Remember that our x-spin vectors were ⟨0, 1⟩ and ⟨1, 0⟩. Because of this it is convenient to talk of them as a “basis” for the vector space, meaning that every vector can be represented as a sum of them, for example ↑z is 1/√2*↑x + 1/√2*↓x. In physics terminology it is thus said that ↑z is a superposition of ↑x and ↓x. But this is just a way of speaking, because we could just as easily define ↑x as a superposition of ↑z and ↓x, it just isn’t as convenient. However, because of this talk of certain states being a “superposition” people sometimes become mislead into thinking that the x-spin properties are somehow overlapping, existing at the same time, ect, none of which is an accurate description. All it means to say that a state is a superposition of some other states is to say that it is not collapsed into one of those states by a measurement, and that is it, no further philosophical implications should be read into it.

And of course all of the particular vector states of the electron, each of which is its own property, can be said to form a family of properties, as any measurement device can determine which state a particular electron is in, given multiple trials (if we can put the electron in that state over an over again we can keep measuring it, and by noting the probabilities by which it collapses into each particular outcome we can determine the initial state before measurement is, to an arbitrarily high degree of accuracy as we take more and more measurements), which is simply a fancy way of saying that they are related to each other (something our mathematical formulation has made intuitively obvious, since they are all represented by simply different vectors in the same vector space).

So, the quantum world is weird because measurement can change the state of what is measured (and because a multiple particle systems can have a state that is not simply the sum of the state of the individual particles, but I’ll leave a discussion of that for another time), but not because it forces us to abandon a common-sense notion of what a property is, as some have claimed.

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4 Comments

  1. So, the quantum world is weird because measurement can change the state of what is measured (and because a multiple particle systems can have a state that is not simply the sum of the state of the individual particles, but I’ll leave a discussion of that for another time), but not because it forces us to abandon a common-sense notion of what a property is, as some have claimed.

    Of course as Richard Feynman noted, this shouldn’t be surprising at all since this is true of the macro-world, albeit in not quite a dramatic way. The way I usually taught it was that if I’m measuring the position of a baseball with a bat I shouldn’t be surprised if the property of momentum changes. All measurements are, in one sense, interactive, and change the system. Some just do so in less significant ways than others.

    The myth QM dispelled was the idea we could converge on some “God’s eye” position where we could measure everything without, in effect, being a part of the system. But that was a silly myth to begin with. I think it was that myth that led to QM seeming so mysterious though.

    Comment by Clark — April 9, 2007 @ 8:17 pm

  2. The baseball bat example doesn’t capture the ‘weirdness’ of non-locality. Imagine that if AFTER you hit the ball, dropping the bat to the left vs the right determines a home-run vs. a pop-fly (not a perfect analogy, but I’m using baseball!) That is the “not simply the sum of the state of the individual particles” part. In macro-world baseball, knowing the state of the ball and of the bat is approximately sufficient for knowing the entire system. In QM, there appears to be LESS than that. But to keep track of how the ‘lessness’ works requires MORE information.

    Comment by Bruce — April 9, 2007 @ 8:53 pm

  3. Also the reason the baseball bat changes the momentum of the ball for a fundamentally different reason than why measuring the x-spin disturbs the z-spin. (Note that the nature of the measurement determines which states it can possibly collapse into, notice also that collape is not affected by how much energy you use in your measurement, you can’t affect it less by measuring it less forcefully.)

    Comment by Peter — April 9, 2007 @ 11:01 pm

  4. Certainly that’s right Peter. (And Feynman of course notes that in his Lectures) But the point Feynman is making isn’t about how it is odd (i.e. how our measuring affects things) merely that we shouldn’t be surprised that it does.

    So certainly non-locality (or the other choices of Bell) are odd. But what people focus in on – that measuring affects – seems to be the thing that is least interesting.

    Comment by Clark — April 10, 2007 @ 6:34 pm


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