You keep ignoring the fact that the dominant eigenstate is derived from nothing but the unitary evolution and the limitations of the observer. This is not a "new theory" or an interpretation of any kind. Since you are not willing to discuss that part your comments regarding the validity of my approach are entirely meaningless. You criticize my work based on the results which are not to your liking, and not with respect to the methods used to obtain these results. So I beg you one last time, let us rationally discuss my arguments, and not what you believe is a valid result or not. If you can show my arguments to be false beyond any doubt, based on the arguments that I use in my blog, or alternatively, if you can point out any assumptions that are arbitrary or not well founded I will accept your statement. But not like this. If you claim to be a rationalist then this is the way to go.
Any other takers out there who are willing to really discuss the matter without dismissing it first?
And just for the record, this has absolutely nothing to do with Bohmian mechanics. There is no extra structure that contains the real outcomes before measurement or any such thing. The only common point is the single reality.
Furthermore, your quote of page 11 leaves out an important fact. Namely that the switching occurs only for the very short time history where the dominant eigenstates interact and stabilizes for the long term, meaning within a few scattering events of which you probably experience billions every second. There is absolutely no way for you to switch between dominant eigenstates with different memories regarding actual macroscopic events.
This would be a lot simpler if you weren't avoiding my questions. I have asked you whether you have understood and accept the derivation of the dominant eigenstate as the best possible description of the state of a system that the observer is part of. I have also asked if you have read my blog from the beginning, because I need to know where your confusion about what I am saying comes from.
The Stern Gerlach experiment goes like this in my theory: The superposition of the spins of the silver atoms must be collapsed already at the moment the beam splits up, because a much later collapse would create a continuous position distribution. That also means a Copenhagen-like act of observation cannot happen any later, specifically not at a screen. This is a good indication that not observation itself forces the silver atoms to localize but something else, that relates to observation but is not the act of looking at it.
In the system that contains the experiment and the observer, the observer would always "see" a state that belongs to the dominant eigenstate of the objective state operator of that system. It doesn't really matter if in that system the observer is entangled with the spin state or not. As soon as you apply the field to separate the silver atoms you also create an energy difference (which is also flight time dependent and scans through a rather large range of possible resonant frequencies). The photons in the environment that are out of the observer's direct observation and unknown to him begin to interact with the two spin states, and some do in a way that creates spin flips, with absorption and stimulated emission, or just shake the atom a little bit. The sum of these interactions can create a total unitary evolution that creates two possible eigenvectors of the state operator, one containing each spin z-eigenstate and a probability for each to be the dominant eigenstate that goes conform with the Born rule. That includes the assumption that the photon states from the environment are entirely unknown. The scattering process I give in my blog shows that such a process is possible and has the right outcome. The dominant eigenstate of the system containing the observer is then the best description of reality that this observer can come up with. Or in other words, he sees either spin up or down and their trajectories.
If you accept the fact that an internal observer can only ever know the dominant eigenstate then state jumps with unknown/random outcome are a necessary consequence. That the statistics of those jumps is the Born rule for events that involve unknown photons is also a direct consequence. And all that follows just from unitary evolution of the global state and the constraints by locality and unitarity on the observer. So please tell me which of the derived steps you do not accept, so that we can focus on it. And please point me to exactly where in the blog the offending statement is.
There must be something that you have fundamentally misunderstood. I will try to clear up some aspects that I think may cause this confusion.
First of all, the scattering processes presented in the paper are very generic to demonstrate the range of possible processes. The blog contains a specific realization which you may find closer to known physical processes.
Let me explain in detail again what this section is about, maybe this will help to overcome our misunderstanding. A photon scatters on a single qubit. The photon and the qubit each bring in a two dimensional state space and the scattering process is unitary and agrees with conservation laws. The state of the qubit before the interaction is known, the state of the photon is external to the observer's system and therefore entirely unknown, and it is independent of the state of the qubit.
The result of the scattering process is traced over the external outgoing photon states to get a local objective state operator. You then write I apply the Born rule, but that's really exactly what I don't do. I use the earlier derived fact that a local observer can only reconstruct the eigenstate with the greatest eigenvalue. This will result in getting either the qubit's |0> or |1> state.
In order to get the exact probability distribution of these outcomes you have to assume exactly nothing about the state of the photon, because it is entirely unknown. If you assume nothing then all polarizations are equally likely, and you get an SU(2) invariant distribution of the coefficients. That's all. There are no assumptions whatsoever about the generation of the photons, them being thermal or anything. Just that all polarizations are equally likely. This is a very natural assumption and hard to argue against. The result in then not only the Born rule but also an orthogonal basis which the outcomes belong to.
So if you accept the derivation that the dominant eigensubspace is the relevant state description for a local internal observer and you accept that the state of the incoming photons is not known, then the Born rule follows for certain scattering processes. If you use precisely the process described in my blog is up to you. It merely stands for a class of processes that all result in the Born rule.
You don't need any modification of quantum mechanics for that. Why do you think you would? Also, this is not just a random combination of algebraic conditions and random distributions. Th assumption about the state distribution of the photon is the only valid assumption if you don't want to single out a specific photon polarization basis. And all the results are consequences of local observation and unitary interactions.
Have you worked through my blog posts from the beginning in the meantime? I ask because I was hoping that they describe all this very clearly. Please let me know if you disagree with how the internal observer reconstructs the quantum state, because I think that's the problem here.
Your question is absolutely valid and also important. In fact, most of what I write in my paper and the blog is about answering precisely this.
My observer is well defined, as a mechanism that is part of a quantum system and who interacts with the quantum system to gather information about it. He is limited by the locality of interaction and the unitary nature of the evolution. I imagine the observer to be a physicist, who tries to describe the universe mathematically, based on what he sees. But that is only a trick in order to have a mathematical formulation of the subjective view. The observer is prototypical for any mechanism that tries to create a model of his surrounding. This approach is very different from modeling cognitive mechanisms, and it's also much more general. The information restriction is so fundamental that you can talk about his subjective reconstruction of what is going on as local subjective reality, as everyone has to share it.
The meaning of the dominant eigensubspace is then derived from this assumption. Specifically, I am able to identify a non-trivial transformation on the objective density operator of the observer's subsystem that he cannot gain any knowledge about. This transformation creates a class of equivalent representations that are all equally valid descriptions which the observer could use for making a model of his environment (and himself). The arbitrariness of the representation connected with this reconstruction however forces him to reduce his state description to something more elementary, something that all equivalent descriptions have in common. And that turns out to be the dominant eigensubspace as his best option. This point is very important, and the derivation I provide in the blog is rigorous and detailed. The result is that the subjective reality as reconstructed by any observer like this evolves unitarily if the greatest eigenvalue does not intersect with other eigenvalues (the observer himself cannot know the value of the eigenvalues either) or discontinuous as a formerly smaller eigenvalue intersects with the greatest one to become the new dominant eigenvalue. This requires an interaction with a part of the system that is not contained in the objective local state description, like an incoming photon.
This approach also has the advantage that you don't have to actually model the observer. You still know what information is available to him. That is why the observer does not even have to be part of the system that you want to "subjectify". You already know how he would describe it. Specifically, you don't have to consider any kind of entanglement between observer states and observed states. The dominant eigensubspace is a valid description of every system that the describing entity is part of and that contains everything the observer is directly interacting with. If you want to get quantum jumps you also need an external inaccessible environment.
Summarizing, there's no need to postulate the ontology or relevance of the dominant eigensubspace. I was very careful to only make assumptions that are transparent and to derive everything from there. Specifically I am not adopting any definition or terminology from interpretations of quantum theory.
You really come up with tricky questions, good :-). I think there are several ways to understand your questions and I am not sure which one was intended, so I'll make a few assumptions about what you mean.
First, an event is a nonlinear jump in the time evolution of the subjectively perceived state. The objective global evolution is still unitary and linear however. In between the perceived nonlinear evolution events you have ordinary unitary evolution, even subjectively. So I assume you mean the subjective states psi1(t) and psi2(t). The answer is then that in general superpositions are not valid subjective evolutions anymore. You can still use linearity piecewise between the events, but the events themselves don't mix. There are exceptions, when both events happen at the same time and the output is compatible, as in can be interpreted as having measured an subspace instead of a single state, which requires mutual orthogonality. So in other words: In general there is no global state that would locally produce a superposition if there are nonlinear local events.
However if you mean that psi1 and psi2 are the global states that produce a list of events a,b,c and A,B,C respectively and you add up those, then the locally reconstructed state evolution will get complicated. If you add with coefficients psi(t) = c1 psi1(t) + c2 psi2(t) then you will get the event sequence a,b,c for |c1|>>|c2| and the sequence A,B,C for |c2|>>|c1|. What happens in between depends on the actual states and how their reduced state eigenspaces interact. You may see an interleaved mix of events, some events may disappear or you may see a brand new event not there before.
I hope this answers your questions.
I see it exactly like you. I too see the overwhelming number of theories that usually make more or less well hidden mistakes. I too know the usual confusions regarding the meaning of density matrices, the fallacies of circular arguments and all the back doors for the Born rule. And it is exactly what drives me to deliver something that is better and does not have to rely on almost esoteric concepts to explain the results of quantum measurements.
So I guarantee you that this is very well thought out. I have worked on this very publication for 4 years. I flipped the methods and results over and over again, looked for loopholes or logical flaws, tried to improve the argumentation. And now I am finally confident enough to discuss it with other physicists.
Unfortunately, you are not the only physicist that has developed an understandable skepticism regarding claims like I make. This makes it very hard for me to find someone who does exactly what you describe as being hard work, thinking the whole thing through. I'm in desperate need of someone to really look into the details and follow my argument carefully, because that is required to understand what I am saying. All answers that I can give you will be entirely out of context and probably start to look silly at some point, but I will still try.
I do promise that if you take the time to read the blog (leave the paper for later) carefully, you will find that I'm not a smuggler and that I am very careful with deduction and logic.
To answer your questions, first of all it is important that the observer's real state and the state that he assumes to be in are two different things. The objective observer state is the usual state according to unitary quantum theory, described by a density operator, or as I prefer to call them, state operator. There is no statistical interpretation associated with that operator, it's just the best possible description of a subsystem state. The observer does not know this state however, if he is part of the system that this state belongs to. And that is the key result and carefully derived: The observer can only know the eigenstate of the density operator with the greatest eigenvalue. Note that I'm not talking about eigenstates of measurement operators. The other eigensubspaces of the density operator still exist objectively, the observer just doesn't know about them. You could say that the "dominant" eigenstate defines the reality for the observer. The others are just not observable, or reconstructable from the dynamic evolution.
Once you understand this limitation of the observer, it follows easily that an evolution that changes the eigenvalues of the density operator can change their order too. So the dominant eigenstate can suddenly switch from one to another, like a jump in the state description. This jump is determined by external interactions, i.e. interactions of the system the observer describes with inaccessible parts of the universe. An incoming photon could be such an event, and in fact I can show that the information contained in the polarization state of an incoming photon is the source of the random state collapse that generates the Born rule. The process that creates this outcome is fully deterministic though and can be formulated, which I do in my blog and the paper. The randomness just comes from the unknown state of the unobserved but interacting photon.
So as you can see this is fundamentally different from MWI, and it is also much more precise about the mechanism of the state reduction and the source of the randomness. And the born rule follows naturally. No decision theory and artificial assumptions about state robustness, preferred basis or anything like that. Just a natural process that delivers an event with a probability measurable by counting events.
Your last question about the environment being classical is a very good one. I do not model the environment to be classical, in fact there is no assumption about it other than that it belongs to a greater quantum system and that it is not part of the system that the observer wants to describe. There are also no restrictions about anything being in a superposition. That problem resolves itself because the state described by the observer turns out to be a pure state of the local system, always. So even if you assume some kind of superposition of these events, you will always get a single outcome. The scattering process in fact has the property of sending superpositions to different eigensubspaces of the state operator, so that it cleans up everything and makes it more classical, just like the measurement postulate would.
I know I am demanding a lot here, but I really think you will not regret spending time on this. Let me know what else I can explain.
I think it will be helpful if I briefly describe what my approach to understanding quantum theory is, so that you can put my statements in the correct context.
I assume a minimal set of postulates, namely that the universe has a quantum state and that this state evolves unitarity, generated by the strictly local interactions. The usual state space is assumed. Specifically, there is no measurement postulate or any other postulates about probability measures or anything like that.
Then I go on to define an observer as a mechanism within the quantum universe that is realized locally and gathers information about the universe by interacting with it.
With this setup I am able to show that an observer is unable to reconstruct the (objective) density operator of a subsystem that he is part of himself. Instead he is limited to finding the eigenvector belonging to the greatest eigenvalue of this density operator. It is then shown that the measurement postulate follows as the observer's description of the universe, specifically for certain processes that evolve the density operator in a way that changes the order of the eigensubspaces sorted by their corresponding eigenvalues.
That is really all. There are no extra assumptions whatsoever. So if the derivation is correct then the measurement postulate is already contained in the unitary structure (and the light cone structure) of quantum theory.
Thank you for your feedback Mitchell,
I'm afraid you have not understood the paper correctly. First, if a system is in a superposition depends on the basis you use to expand it, it's not a physical property but one of description. The mechanism of branching is actually derived, and it doesn't come from superpositions but from eigenstates of the tensor factor space description that an observer is unable to reconstruct. The branching is also perfectly deterministic.
I think your best option to understand how the dominance of one branch and the non-reality of the others emerges from the internal observation of unitary evolution is to work through my blog posts. I try to explain precisely where everything comes from and why it has to follow. The blog is also more comprehensible than the paper, which I will have to revise at some point. So please see if you can more make sense of it from my blog, and let me know if you still understand what I'm trying to say there. Unfortunately the precise argument is too long to present here in all detail.
Have you checked out the posts at my blog? I don't know about your background, but maybe you will find them helpful. If you would like to have a more accessible break down then I can write something here too. In any case, thank you for your interest, highly appreciated!
I'm a theoretical physicist from Germany. My work is mostly about the foundations of quantum theory, but also information theory and non-commutative geometry. Currently I'm working as head of research in a private company.
As a physicist I have been confronted with all sorts of (semi-) esoteric views about quantum theory and its interpretation, and my own lack of a better understanding got me started to explore the fundamental questions related to understanding quantum theory on a rational basis. I believe that all mainstream interpretations have issues and that the real answer is a rigorous theory of quantum measurement. On my blog at http://aquantumoftheory.wordpress.com I argue that quantum theory does not have to be interpreted and I propose a rational alternative to interpretation. This is also the main reason I came here, to discuss my results with other rationalists to see if they are indeed satisfying. So your feedback is very welcome!
Other interests of mine include cognitive psychology, music (both active and passive), cooking and photography. Science in general and the philosophy of science, at least the more rational parts, are also interests of mine.