Ever since I was a a highschool student and certainly as a physics student I have been wrestling with what some people call "the interpretation of quantum mechanics". (I have put this term into quotation mark because I think it is probably a misnomer. What people may mean by it, often subconciously, is to understand quantum mechanics in terms of classical physics. This can bring you into a lot of trouble trying to get a clearer understanding, and I speak from experience here. What we really should do is to understand classical physics in terms of the more fundamental theory, quantum mechanics.) Although I do not work in physics anymore, the question has never completely left me alone. The following article summarizes my current thinking using the thought experiment of Schrödingers cat. The best sources of theoretical physics available on the internet have gone into it. Let me name physicists like Nima Arkina-Hamed, Lubos Motl, Sidney Colement and David Mermin.
Probably the most famous thought experiment about quantum mechanics is Schrödinger's cat. A cat is enclosed in a box for a certain amount of time together with a machine which may or may not kill it depending on the result of a quantum mechanical experiment. In the original version by Schrödinger it is a radioactive substance and for simplicity he imagines that the radiactive substance is so tiny, that with 50% probability one atom will decay within an hour and by doing so will initiate the killing of the cat. After one hour the box is opened and it is checked whether the cat is alive or not.
Schrödinger invented his thought experiment to show that quantum mechanics when applied to macroscopic objects would lead to paradoxical situations. Why did he think so? Without changing the character of the experiment let us imagine that the killing apparatus is connected to a measurement device which measures the spin of an electron in the z direction. If the spin is positive, the cat will be killed, while if it is negative, the apparatus will do nothing. To get the 50% probability we imagine that before the experiment started the electron spin was measured in the x direction with a + result and afterwards the electron was isolated from the environment. This means the electron is in a state: \[ \frac{1}{2} \ket{up} + \frac{1}{2} \ket{down} \] After the box is closed, the killing machine will do its work and develop the state into: \[ \frac{1}{\sqrt2} ( \ket{up} \ket{ cat \; alive}+ \ket{down}\ket{ cat \;dead } )\] So what we arrived at is a superposition of a state of a dead cat and an alive cat. At that point Schrödinger concluded, that there is a paradox. He would say that it is obvious from our experience, that a cat cannot be at the same time dead and alive. This may just be true for very small things like atoms, which we do not have direct experience with. So has Schrödinger succeeded and shown, that the rules of quantum mechanics, when applied to macroscopic objects, lead to absurd results and the macroscopic world cannot be described by the rules of quantum mechanics just like that? Not so fast. As physicists we should remember that we should judge the success of a physical theory by its ability to predict experimental result. And the only way to extract these predictions is to follow what the theory tells us to do. In other words a state vector in quantum mechanics is just a tool to make predictions. Based on the setup of the experiment when we open the box we should see either a dead cat or an alive cat with 50% probability respectively. How do we get this result by actually applying the formalism of quantum mechanics to our state vector? Perhaps the easiest way is to calculate the overlap the state vector with the subspace where the cat is dead and then take the absolute value squared of that vector: \[ \bra{ any \;elecron \;spin } \bra{ cat\; alive} \frac{1}{\sqrt2} ( \ket{up} \ket{ cat \; alive}+ \ket{down}\ket{ cat \; dead} ) \] Since the states for alive cat and dead cat are orthogonal with very high accuracy, the projection gives \( \frac{1}{\sqrt2} \) and so the probability is \( \frac{1}{2} \) as expected. (Also the projection will only give a none-zero result, when we chose the electron spin as spin up, as expected from the experimental setup.)Let me emphasize, that measuring a quantum system always contains of two steps. This is in my opinion often not enough emphasized in many treatments. To see this, let us leave the poor cat out of the experiment and emphasize the role of the detector. This becomes an ordinary Stern-Gerlach experiment and the quantum state after the detector interacted with the system is \[ \frac{1}{\sqrt2} ( \ket{up} \ket{ detector \; flashing \; up }+ \ket{down}\ket{ detector \; flashing \;down } ) \] The calculation of the probability amplitude for measuring spin up is calculated as above and the result is again \( \frac{1}{2} \). The reason why we always need this interaction is that we as humans are big and our senses cannot directly interact with this system. Our eyes cannot observe single photons and the eardrum in our ears cannot hear individual atoms hitting against is. So the interaction is needed for amplification to get a signal, which we can directly observe such as a click of a detector, a flash of light, a bright spot on a a screen or fotographic plate and so on. It is important to note that this amplification involves a choice of the experimenter what he wants to measure. In our example we can make a decision whether we want to measure the spin in the x or in the z direction for example or any other direction. Now the interaction can be entirely described by the laws of quantum mechanics itself, e.g. the Schrödinger equation. This is usually not done and not necessary because we care about the system itself and we assume that the detector is working correctly in the sense it amplifies the property we want to measure. So we assume a "collapse of the wave function" to one of the eigenvalues of the measured quantity and calculate the probabilities of each eigenvalue to occur as the absolute value of the measured quantity. This is called the Born rule. But we can (and in fact things at least roughly following this idea are studied and done under the name of weak measurements) also "measure" our single atom by letting it interact with a molecule of 50 atoms flying by. Then measurement becomes a quantum mechanical process and using the Born rule just on the system alone will not give correct results. In the end however there always needs to be an amplification to the macroscopic level. The second step of measurement is observation. Observation means that an observer notes one particular outcome of the experiment. This may sound tautological. I will talk more about the role of the observer in a bit. Now if we look at the daily experience of humans beings quantum measurements are rare. They are done by physicists in the laboratory and the system to be measured (e.g. a single atom) needs to be isolated from the environment usually by cooling and putting it into a vaccuum shielding it from light and other radiation. (Now you may say that the world is quantum mechanical and therefore all measurement is quantum measurement, even just measuring the length of the cupboard in your living room with a measuring rod. And indeed in a sense you are exactly right about this. What I mean by quantum measurement is a measurement which reveals the quantum nature of our world directly. Most ordinary measurements involve so many particles, that the quantum properties of these particles simply avarage out.) On the other hand the two elements of quantum measurement, which is interaction and observation occur all the time. Interaction happens between the air molecules in your room, by these air molecules hitting the wall and the chair you are sitting on and so on and so on. Interaction between atoms and molecules and light and everything around you is so common that it requires hard work of experimental physicists to limit and control it. In a certain sense the world around us is constantly measuring itself through interaction. This may seem astonishing because it only consists of those fuzzy quantum particles. What is meant by this that the probability amplitudes of quantum mechanics which make it possible for the genuin quantum phenomenon of interference to occur, are constantly transformed by self-interaction into classcial probabilities. This process is called decoherence. Observation is also occurring all the time, as long as you have a conscious experiments and in fact we not only subscribe it to humans, but also to dogs and elephants and other higher life forms. So the only difference between performing a quantum measurement and waking up and opening your eyes in the morning is, that in the latter case neither you nor anybody else is controlling the quantum interactions leading to your observations. This means you observe your entire environment as a quantum system and not just isolated parts of it amplified by a detector.
Now we may already end the article here by concluding that quantum mechanics gives us the right prediction for the experiment. What more can you expected from a successful physical theory? And quantum mechanical certainly is very successful theory and viewed as a framework at the heart of all fundamental physical theories still today, including quantum field theory and string theory. If you prefer a more practical view, it is often said, that quantum mechanics enables one third of the wordwide GDP. But if things are so simple, why has Schrödinger as one of the founders of quantum mechanics, other great physicists like Einstein (Einsteins view deserves certainly an article of its own) and also other physicists and people interested in physics including myself problems. I think the reason is twofold. On the one hand people expect more from a physical theory than making successful predictions for experiments. They expect a physical theory to give inside to what reality is. Secondly quantum mechanics seems to give a special role to the observer. It seems that without the observer Schroedingers cat will stay forever in this superposition between dead and alive. Or in other words without an observer nothing seems to be ever happening in our world! So has physics, our hardest and most exact and mathematical science become subjective? We can try to eloaborate on this question by introducing an extension of the Schrödinger's cat experiment called Wigner's friend.
Here we put an observer called Wigner's friend into the box while another observer, whom we may call Wigner, stays outside to open it. After the machine has potentially done its work, but before the box is opened, Wigner's friend sees either the state \(\ket{alive\; cat}\) or the state \(\ket{dead\; cat} \)with 50% probability but no superposition. (We omitted the electron spin in the state description.) The usual language is that because of the observation for Wigner's friend the wave function has “collapsed”. For Wigner instead the collapse did not take place as he did not open the box yet and the state is a superposition of a dead cat together with an unhappy friend on the one hand (Wigner's friend likes cats) and an alive cat together with a happy friend on the other hand. At this stage Wigners quantum state will look like this: \[ \frac{1}{\sqrt2} ( \ket{up} \ket{cat \; alive} \ket{Wigners \;friend \;happy} + \ket{down}\ket{cat \;dead} \ket{Wigners \;friend \;unhappy} ) \] So we have to indeed conclude that quantum mechanics is subjective. The "collapse of the wave function" has happened for Wigners friend, but not for Wigner. However we should try to not overinterpret this and see it as something overly mystical. Quantum mechanics is a fundamentally probabilistic theory. That means the probabilities are not a result of our inability to accurately calculate. Instead they are at the deepest level of reality. With Einstein you may doubt that this is the case but nobody has found any indication for a deeper layer of reality going beyond quantum mechanics. And if you insist, as Einstein did, on "hidden variables", which make the underlying theory below quantum mechanics deterministic, there is an own set of problems associated to it. Much of this discussion crystalizes around the so called EPR paradox. But this is the topic of another article. What I want to say here is that if you have a probabilistic theory, then you should not be surprised that the observer makes an appearance at the fundamental level of this theory. Your immediate reaction may be to ask who counts as an observer, a human, a dog, an insect, maybe a smart computer? My answer to this would be that the observer is a irreducable concept of the theory. You can compare this sitation to mathematics. In number theory you never explicitly define what a number is. Instead it is implicitly defined by the Peano axioms. The same applied to concepts like point or line in geometry. In any theory at some point the explicit definition of concepts by even simpler concepts has to stop, when you arrive at the fundamental concepts, a theory talks about. This is what I mean when I say that the concept of an observer cannot be further reduced in quantum mechanics. Let me comment on what I mean by saying that quantum mechanics is subjective. I think we all have an intuitive picture of the world that reality is objective. My favourite sports team either won yesterday's match or it did not, even if I was not in the stadium and did not get the result yet via media. Of course quantum mechanics as our best theory of the world does reflect that. Yes, the unitary evolution of quantum mechanics is time reversal symmetric so I might try to undo the killing of the cat by applying the right unitary evolution. But the rules of thermodynamics exist in quantum mechanics just as in classcial physics and we do not have the power to control the \( 10^{23} \) atoms of a cat (not even of a bacterium). So I can treat the Schroedingers cat state as two alternatives with classical probabilities. The complex phases, which distinguish quantum mechanical probability amplitudes from classical probabilities, do not matter anymore. The alternatives cannot interfere, because they have imprinted themselves into different states of the environment, which can be regarded with very high accuracy as orthogonal. So quantum mechanics is subjective in the same way that classical probability theory. Whenever your state of knowledge changes, you have to adjust your prior probabilities. That is what we call the collapse of the wave function in quantum mechanics.
In the past the main flaw I saw in quantum mechanics was that nowhere in the theory there is a place, where probability amplitudes are turned into actual outcomes. I would learn that decoherence should solve the problem because it is able to turn probability amplitudes into classcial probabilities as I have sketched above. But I told to myself that this was just cheating and the actual problem was not addressed. A probability for something to happen is good and well but where is the place in nature, where an outcome is generated. I was right about the fact, that there is indeed no such place, no such mechanism in quantum mechanics. But is this a problem? Here I found a lecture of David Mermin interesting. He made exactly the same observation but he compared it with a situation in classical physical which he calls "the problem of the now". In classical physics time is usually represented as a real number, as an axis in a coordinate system, where the other axis are spacial dimensions. Now everybody experiences reality as a continuous sequence of moments. Every bit of experience is linked to a particular moment. But in classical physics this moment is not represented. When you draw a diagram of a body moving in space as a function of time, all moments of time exist in parallel. There is no pointer in the diagram, which is moving accross the time axis and is telling you what the current moment is. While this is true and people like Einstein seems to have thought about this, nobody finds classical physics incomplete for this reason. So, as David Mermin suggests, we should perhaps not be disturbed by the fact, that our experience of things happening is not reflected in quantum mechanics, just as we are not disturbed by classical physics not explaing to us, why we experience time as a sequence of moments, which are happening "now".
To be continued...