Many things can “happen” inside a quantum wave function, of course, including everything that actually does happen — formation of galaxies, origin of life, Lady Gaga concerts, you name it. But given a certain quantum wave function, what actual is happening inside it?
A surprisingly hard problem! Basically because, unlike in classical mechanics, in quantum mechanics the wave function describes superpositions of different possible measurement outcomes. And you can easily cook up situations where a single wave function can be written in many different ways as superpositions of different things. Indeed, it’s inevitable; a humble quantum spin can be written as a superposition of “spinning clockwise” or “spinning counterclockwise” with respect to the z-axis, but it can equally well be written as a superposition of similar behavior with respect to the z-axis, or indeed any axis at all. Which one is “really happening”?
Answer: none of them is “really happening” as opposed to any of the others. The possible measurement outcomes (in this case, spinning clockwise or counterclockwise with respect to some chosen axis) only become “real” when you actually measure the thing. Put more objectively: when the quantum system interacts with a large number of degrees of freedom, becomes entangled with them, and decoherence occurs. But the perfectly general and rigorous picture of all that process is still not completely developed.
So to get some intuition, let’s start with the simplest possible version of the problem: what happens inside a wave function (describing “system” but also “measurement device” and really, the whole universe) that is completely stationary? I.e., what dynamically processes are occurring while the wave function isn’t changing at all?
You’re first guess here — nothing at all “happens” inside a wave function that doesn’t evolve with time — is completely correct. That’s what I explain in the video above, of a talk I gave at the Philosophy of Cosmology workshop in Tenerife. The talk is based on my recent paper with Kim Boddy and Jason Pollack.
Surprisingly, this claim — “nothing is happening if the quantum state isn’t changing with time” — manages to be controversial! People have this idea that a time-independent quantum state has a rich inner life, with civilizations rising and falling within even though the state is literally exactly the same at every moment in time. I’m not precisely sure why. It would be more understandable if that belief got you something good, like an answer to some pressing cosmological problem. But it’s the opposite — believing that all sorts of things are happening inside a time-independent state creates cosmological problems, in particular the Boltzmann Brain problem, where conscious observers keep popping into existence in empty space. So we’re in the funny situation where believing the correct thing — that nothing is happening when the quantum state isn’t changing — solves a problem, and yet some people prefer to believe the incorrect thing, even though that creates problems for them.
Quantum mechanics is a funny thing.
Ravi Ivaturi and Mr. Carroll,
Just because a particular mathematical equation in the one to use in
order to solve a problem, does not mean we should take every aspect of
it completely seriously. Not every aspect corresponds to something
that actually exists.
For example, suppose I told you that the length of a rectangle is two
more than its width, and that the area of the rectangle is 8, and your
task was to solve for the width, x.
Then, we have the equation x(x+2) = 8 and its solutions are x = 2 or x
= -4. Does that equation predict the existence of two solutions?
Does it predict the existence of this novel concept of negative
length? No. x = -4 is just a byproduct of the mathematical
formalism, a byproduct of the mathematical/calculational tool that you
used to solve the problem.
So if one claims that the mathematical formalism of QM predicts the
existence of all these other worlds, then perhaps we should treat the
mathematics as merely a calculational tool used to solve some problems
and not take things too seriously or too literally.
For example, if I just gave you the metric for the Schwarzschild black
hole, you’d be able to see that there appears to be a singularity at
the event horizon. Just using that concept and equation, you would
think that, indeed, a singularity exists at the event horizon.
However, once you introduce the notion of curvature and curvature
invariants, you’d be able to see that no singularity exists there. So
perhaps, if you think QM leads to the existence of many worlds, we are
missing key concepts from a deeper underlying theory that prevents
those ridiculous things from coming up? Perhaps instead of thinking
about ridiculous things like many worlds everytime a measurement takes
place, we need to understand the fundamental theory of space-time and
quantum gravity. And that, I’m afraid, can only be done with
experiment, since our human brains might not be intelligent enough to
figure out what’s going on at 10^-35m ! 🙂 Or perhaps someone
brilliant will come along.
What we really need is experiment, predictions that can be checked by
performing these experiments, and falsifiability. That’s important
for the scientific method.
“Then, we have the equation x(x+2) = 8 and its solutions are x = 2 or x
= -4. Does that equation predict the existence of two solutions?
Does it predict the existence of this novel concept of negative
length? No. x = -4 is just a byproduct of the mathematical
formalism, a byproduct of the mathematical/calculational tool that you
used to solve the problem.”
x=-4 is a valid solution. The ruler is -4 away from the origin and its adjacent side is 2 more the same distance from the origin, which is -2.
I wouldn’t agree that a fundamental physical theory has mathematical semantics that have no physical interpretation. If it’s an accurate theory, then there are sensible physical interpretations that exist. They just may not be experimentally verifiable.
x = -4 is a valid solution if we are dealing with the field of real
numbers, but the concept of length is associated with the field of
positive real numbers, so in that case x = -4 is not a valid solution.
By understanding that x’s interpretation is that of the length of a
geometric object, we come to the understanding that we have to throw
out one of the solutions.
Likewise, perhaps when we figure out what the best interpretation is
of things like the wavefunction, or whatever important quantity comes
along in our quest for the fundamental theory of nature, we will need
to throw out these sorts of byproducts of our mathematical tools used
to investigate nature.
“I wouldn?t agree that a fundamental physical theory has mathematical
semantics that have no physical interpretation.”
There’s no reason to believe that. It could also just be that the
mathematical theory describing the fundamental laws has mathematical
baggage that doesn’t correspond to nature. If we view the work of
physics as the development of mathematical models that describe nature
with better and better accuracy, then there’s no reason to think that
the mathematics behind the fundamental theory is anything more than
another mathematical model. And with mathematical models comes the
possibility of mathematical baggage that has nothing to do with actual
physical reality. But who knows, maybe you are correct.
You’re artificially demanding that length is strictly a positive number. The fact is a legitimate interpretation exists and you discard it only because it’s inconvenient. Allowing length to be positive and negative allows you to define orientation, it’s actually more descriptive than your restriction to the absolute value of the coordinate positions that define the edges of this rectangle.
I agree with your other points, but as long as valid interpretations exist, like the one above, you can’t justifiably claim it’s all “baggage.” You have to discount the interpretations first. You can’t do that for the rectangle as there are clearly semantics where the full range of real numbers make sense. It’s a lot harder to do this for quantum mechanics.
More to the point, the axioms of quantum mechanics establish a specific correspondence between mathematical objects and physical ones. The complete Hilbert space is mapped to by these semantics, so there’s nothing extra here. Many Worlds just has an extra assumption about how the Hilbert space is constructed in the first place, how the probability space arises naturally from a ground-up approach. Maybe this is extra, maybe not, but how the Hilbert space arises from fundamental measurements is something that has to happen. The wavefunctions are not physically what are measured directly.
@Meleny
I think you misunderstand how the MWI can be an interpretation of the mathematical solution to quantum mechanics. In a way it is like the example you give, the equations literally imply that the nature of the quantum world is probabilistic, since the mathematics only deals with probabilities. Then the MWI allows people to think of the probabilities they are dealing with as just a consequence of trying to deal with and describe many worlds branching off of each other or decohering. When an event happens, it was just calculated that the probability of that world being our reality has taken place, instead of there just being one world with probabilistic quantum properties by nature. There is actually nothing in the mathematics of quantum mechanics that says there has to be many worlds.
Personally, I think things are just that way, because of the failure to unify relativity and quantum mechanics. It may not be possible, because relativity breaks down at the speed of light. Also, quantum mechanics has to deal with particles traveling the speed of light. Then that is were the “hidden variables” lie. It may just be that mathematics is not advanced enough to give an accurate description of such a theory. It could take having to deal with zero’s and infinities mathematically in different ways, or it could be that zero’s and infinities in mathematics make something have to be probabilistic by nature. It could also mean that each theory may just need some minor adjustments, or it actually may just be mathematically impossible altogether, making probabilities the best mathematical description of the theory that we will ever get.
Sean has an interesting talk, really loved it, but it is mostly wrong for a reason he hasn’t considered. There are two dimensions of time, each of which uses the other as a state, so the Schrodinger equation, has two parts one which evolves the T1 state in the T2 time, and one that evolves the T2 state in the T1 time. I first discovered this, when I realized that there are two types of Energy, one that is a derivative of T1 and one that is a derivative of T2. If you accept that there are two dimensions of time, that it is obvious that the observer in any one system is the other dimension of time– You actually see this in the Twin paradox, when one twin ages at a different rate than the other. Good job Sean, but you need a better imagination.
Richard M. Kriske, kris0022@umn.edu
One of my favorite interpretations of the Wave Function is John Cramer’s Transactional Hypothesis. When I first read it 20 years ago, it seemed like such an elegant idea. http://en.wikipedia.org/wiki/Transactional_interpretation
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Well senor Kerr, it seems like a bloated belief because it’s extremely rare(nonexistent) and/or outside of our current understanding of physics.
When it’s said that a particle can be in two places at the same time, is there any limit to how far apart these two places can be? While I only have a layman’s, non-mathematical, knowledge of Quantum Mechanics, I assume it’s a function of the Uncertainty Principle involving a trade-off between the total amount of time the particle can occupy two separate locations, and the separation distance between these locations. Would this be correct?