There is a new TV series titled Dark Matter on Apple TV+. It is based on a novel of the same title by Blake Crouch. I have not read the book yet but it is on my reading list. The show’s basic premise is the following: Jason Dessen was once a promising physicist but chose to have a family with love of his life, Daniela over his ambition in quantum physics research 15 years ago. Now he is teaching physics at a college and living an ordinary and peaceful (but somewhat boring) life with his wife Daniela and son Charlie. One night, on his way back home from a bar, he was kidnapped and was thrown into a world that he doesn’t recognize…
The show is quite entertaining. Jason has to go through some (mostly unpleasant) adventures, which reminded me of the 1990’s Sci-Fi TV series Sliders, until he figures out how to control superposition and to select the reality that he wants. (Wouldn’t this be the ultimate dream of all quantum computing scientists?) Contrary to the title, the show’s plots do not have anything to do with dark matter, but the main scientific idea of the show appears to be based on Hugh Everett’s many-worlds interpretation of quantum mechanics just like another recent TV series on Apple TV+, Constellation which I wrote about here. The show Constellation made less physical sense. For example, there was a lack of explanation why two particular realities matter among infinitely many choices of realities as superposition or how an observer experience two different realities against the collapse of wave function. On the other hand, in the show Dark Matter, at least some ideas (brilliant, though not so really scientific or logical) were introduced in an attempt to address the observer effect (collapse of wave function) or a manifestation of superposition.
In here, besides the measurement issue (which is critically important for a viable physical theory) with many world interpretation, I also mentioned that extending the wave nature of matter (consequently, the probabilistic nature of quantum mechanics) to the macroscopic world or to the entire universe is too far-fetched to be even remotely true. I am going to elaborate more on this. But first,
What is a Superposition?
A superposition is actually a mathematical concept. While a superposition is a common terminology in physics, in mathematics, it is usually called a linear combination. To explain a linear combination, I will have to start from vectors. As you learned in high school physics, a vector is a quantity that has both magnitude and direction. So, vectors are often represented by directed line segments i.e. arrows. On the other hand, a quantity that has only magnitude is called a scalar. (A scalar is nothing other than a number.) You also remember that one can add two vectors using a parallelogram or equivalently a triangle. One can also multiply a vector by a scalar. It’s called scalar multiplication. One can stretch or shrink a vector, or reverse the direction of a vector by scalar multiplication. (If you want to brush up about vectors, see here.) It all made sense until higher dimensional spaces came in. For instance, Einstein’s theory of special relativity taught us that the world is not 3-dimensional (as in Newtonian physics) but actually 4-dimensional, called spacetime. Unfortunately, for being 3-dimensional beings our perception is limited to 3-dimensions, so the classical notion of a vector is not adequate to study mathematics or physics in higher dimensions. It turns out, as an alternative and a general definition of a vector, an $n$-dimensional vector can be defined by an $n$-tuple $(a_1, a_2,\cdots, a_n)$. (For details, see here.) This certainly provides a nice framework for studying mathematics and physics in higher dimensional spaces. However, as we advance our knowledge in mathematics and physics, we realized that such a general definition of a vector is still not adequate to deal with new mathematical objects arising in mathematics and physics. This time, mathematicians got smarter. They chose not to define an individual vector. Noting that the set of classical vectors with vector addition and scalar multiplication satisfies certain properties associated with the operations, they defined a vector space by any set with addition and scalar multiplication satisfying those properties as axioms. See here for details on the definition of a vector space. Once we have a vector space, any element belonging to the vector space is called a vector. This allows us to identify certain objects arising not only in mathematics but also in physics (for example, quantum mechanics) and in engineering (for example, signal processing) as vectors. From classical sense, it is inconceivable to see them as vectors as we will see later. Let $V$ be a vector space. Let $v_1, v_2\in V$, i.e. $v_1, v_2$ be vectors and $c_1, c_2$ scalars. Since $V$ is closed under vector addition and scalar multiplication, we see that $c_1v_1+c_2v_2\in V$. The expression $c_1v_1+c_2v_2$ is called a linear combination or a superposition of two vectors $v_1$ and $v_2$.
By the way, I must admit that what I described above may not be (or more likely, is probably not) how the notion of a vector has evolved in mathematics. In other words, I made things up to emphasize the need for an evolution of the notion of a vector in accordance with the progress of modern physics. So, what is the real story? I don’t know. Regrettably, I am not so privy to history of vectors. But it’s not really important here.
Ok, Then How Does a Superposition Come into Quantum Mechanics?
A quick answer is the famous Schrödinger equation $$i\hbar \frac{\partial\psi(\vec{x},t)}{\partial t}=H\psi(\vec{x},t)$$ where $H=-\frac{\hbar^2}{2m}\nabla^2+V$ is a Hamiltonian. If you have no idea what those symbols mean, that’s okay. You don’t actually have to know them. All you have to know is that Schrödinger equation tells us how particles such as electrons behave and mathematically, it is a second-order linear partial differential equation. It is quite mouthful but the most important keyword here is linear. What that means is that if $\psi_1$ and $\psi_2$ are solutions of Schrödinger equation, so is $c_1\psi_1+c_2\psi_2$, i.e. a superposition of $\psi_1$ and $\psi_2$. This means that the set of all solutions of Schrödinger equation form a vector space! Furthermore, it becomes in general an infinite dimensional vector space called a Hilbert space. So, in quantum mechanics, functions $\psi$ as solutions of Schrödinger equation are vectors. Before continue, what is $\psi$? $\psi$ is so called a state function or a wave function. It represents a certain physical state of a particle, hence called a state function. But then why is it also called a wave function? Because it is literally a (complex-valued) wave. You remember hearing about the wave-particle duality of matter even in high school physics, right? Just like light, particles such as electrons also act like waves. So, physically it makes sense to use a wave function to represent a state of a particle but it is also easier to use a wave function to model quantum mechanics. I don’t mean to delve too much into quantum mechanics here as this article is not intended to be a lecture note on quantum mechanics. But I would like to show you how a wave function naturally entered into building quantum mechanics. First, the following complex-valued plane wave \begin{equation}\label{eq:debroglie}\psi(x,t)=Ae^{i(kx-\omega t)}\end{equation} was well-known to physicists even long before the birth of quantum mechanics. For the sake of simplicity, I am considering only one-dimensional case. The plane wave has quantities that characterize a wave: the wave number $k$ and the frequency $\omega$. Reasonably enough to guess, physicists might have thought of using the wave function to model a particle like an electron. A lot of time, modeling means writing an ordinary or a partial differential equation that describes a physical phenomenon. Differentiating \eqref{eq:debroglie} with respect to $t$ yields $$i\hbar\frac{\partial\psi}{\partial t}=\hbar\omega\psi=E\psi$$ and differentiating \eqref{eq:debroglie} twice with respect to $x$ yields $$-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}=\frac{\hbar^2 k^2}{2m}\psi=\frac{p^2}{2m}\psi$$ For a free particle, kinetic energy is the only energy involved, hence we obtain free Schrödinger equation $$i\hbar\frac{\partial\psi}{\partial t}=-\hbar^2\frac{\partial^2\psi}{\partial x^2}$$ With the dispersion relation $\omega=ck$ for light, one can also easily show that \eqref{eq:debroglie} satisfies the wave equation $$-\frac{1}{c^2}\frac{\partial^2\psi}{\partial t^2}+\frac{\partial^2\psi}{\partial x^2}=0$$ A word of caution here. What I said above is in no way a historical account of how quantum mechanics was actually formulated. I am not privy to history of quantum mechanics either, so I don’t know. I am trying to show what one could have undertaken to mathematically model quantum mechanics in the beginning from an already well-known wave function in electromagnetism. \eqref{eq:debroglie} resembles circular water waves from a single source but apparently that is not how free particles behave, so the plane wave itself is not a mathematical manifestation of a free particle. How do we then interpret the wave function and relate it to a (free) particle? Physicists interpret it as something that represents a physical state of a particle. There can be many different physical states such as bound state and spin state, etc. depending on Hamiltonians. Since wave function itself is not observable (not something we can physically measure, i.e. numbers), it has to be connected to an observable. In general, a superposition state $\psi$ is given by $$\psi=\sum_{i=1}^\infty c_i\psi_i=c_1\psi_1+c_2\psi_2+\cdots$$ where $\psi_i$, $i=1,2,3,\cdots$ are eigenstates. Whenever a measurement is done, $\psi$ collapses to an eigenstate $\psi_i$. But again, we don’t really measure an eigenstate $\psi_i$ but its corresponding eigenvalue (energy) $E_i$ (they satisfy the Schrödinger equation $H\psi_i=E_i\psi_i$). Each eigenstate $\psi_i$ can be normalized, meaning it can be made into satisfying $|\psi_i|^2=\bar\psi_i\psi_i=1$. The quantity $$\mathrm{Pr}(\psi=\psi_i)=\frac{|c_i|^2}{\sum_{i=1}^\infty|c_i|^2}$$ in interpreted as the probability of the superposition $\psi$ being the eigenstate $\psi_i$.
Schrödinger’s Cat
Schrödinger’s cat is a thought experiment used by physicists to explain quantum superposition. Regrettably, often by misunderstanding or on purpose laypersons are misled to believe that quantum superposition is something so out of the world phenomenon. Here is what the thought experiment is about.
Consider a sealed box containing a cat, a flask of poison, a radioactive substance and a Geiger counter.
If the Geiger counter detects a radiation due to the radioactive substance decaying, the flask will be shattered, releasing toxic gas which will kill the cat. Over the course of time, the cat is simultaneously alive and dead. Once you open the box, the state of the cat collapses to either alive or dead.
Here are some problems with the conventional interpretation of Schrödinger’s cat. First, the state of a cat is not a quantum state. So applying the superposition principle of quantum state to the state of a cat is just far-fetched. Second, the cat being simultaneously alive and dead is misleading. Although we can’t observe the cat until we open the box, she is either alive or dead at any moment in time. The cat will never be half-alive and half-dead. The correct description of what’s going on is that the probability of the cat being alive (or dead) is 50% at a certain moment in time. So there is really no mystery after all regarding quantum superposition contrary to what has been portrayed in physics literature and Sci-Fi.
There Is No Universal Wavefunction
To recap, there is quantum superposition due to the wave nature of matter and the fact that the equation of motion (Schrödinger equation) for such matter waves is linear. There is no wave function for the state of the poor cat inside a sealed box and she will never be in any quantum superposition. Likewise, there is no physical justification or reason why there should be universal wavefunction, i.e. the wave function for the whole universe. Hugh Everett’s many-worlds interpretation is based on the premise (existence of universal wavefunction) that is purely speculative. This means that whatever box Jason builds he will never experience quantum superposition of the world i.e. infinite possibilities of realities. To his greatest disappointment, nothing will happen in the box. Regardless, if you don’t think about real physics, I believe you will enjoy the show.
So Does This Mean There Are No Parallel Universes?
No, many-worlds interpretation of quantum mechanics is not the only thing that implies the existence of parallel universes. For example, parallel universes can exist as a consequence of traveling back in time. Is traveling back in time possible? Yes it is theoretically possible to travel back in time using a wormhole as seen in many recent studies by physicists. Besides technical possibilities of traveling back in time, the biggest problem with traveling back in time is causality violation such as the grandfather paradox. The grandfather paradox exists based on the assumption that time is one-dimensional. If time is multi-dimensional, we no longer have grandfather paradox! How time can be multi-dimensional? It can happen due to a Chronology Protection Conjecture (CPC). As for time travel, there are currently four different CPCs.
- The Radical Rewrite Conjecture: One can travel forward and back in time and rewrite the history. (All Hell breaks loose.)
- Novikov’s Consistency Conjecture: The Universe is consistent, so whatever temporal transitions and trips one undertakes, events must conspire in such a way that overall result is consistent.
- Hawking’s CPC: The cosmos works in such a way that time travel is completely and utterly forbidden. This conjecture permits spacewarps/wormholes but forbids timewarps/time machines.
- The Boring Physics Conjecture: There are no wormholes and/or spacewarps. There are no time machines/timewaprs.
I am personally fond of Novikov’s consistency conjecture but I put forth another CPC which is a stronger version of Novikov’s consistency conjecture.
Protect History At All Cost!
Here is Lee’s CPC: It does permit timewarps/time machines but does not permit any violation of causality and any tempering with already recorded history.
A consequence of this CPC is that as soon as any tempering with recorded history occurs, a new timeline must be created in order to prevent history from being rewritten, i.e. a new reality is created separately from the previously existing reality. This new reality shares exactly the same history up until the moment a tempering with history was occurred. From that moment onward, it may turn out to be pretty similar to the previous reality or it may turn out to be completely different.
I am currently writing a book on time travel and more details about this will be discussed there.
