Disclaimer: If I had something well thought through, consistent, not vague and well supported, I would be sending it to Phys.Rev. instead of using it for karma-mining in the Irrationality thread on LW. Also, I don’t know your background in physics, so I will probably either unnecessarily spend some time explaining banalities, or leave something crucial unexplained, or both. And I am not sure how much of what I have written is relevant. But let me try.
The standard formulation of the quantum theory is based on the Hamiltonian formalism. In its classical variant, it relies on the phase space, which is coordinatised by dynamical variables (or observables; the latter term is more frequent in the quantum context). The observables are conventionally divided into pairs of canonical coordinates and momenta. The set of observables is called complete if their values determine the points in the phase space uniquely.
I will distinguish between two notions of state of a physical system. First, the instantaneous state corresponds to a point in the phase space. Such a state evolves, which means that as time passes, the point moves through the phase space along a trajectory. It has sense to say “the system at time t is in instantaneous state s” or “the instantaneous state s corresponds to the set of observables q”. In the quantum mechanics, the instantaneous state is described by state vectors in the Schrödinger picture.
Second, the permanent state is fixed and corresponds to a parametrised curve s=s(t). It has sense to say “the system in the state s corresponds to observable values q(t)”. In quantum mechanics, this is described by the state vectors in the Heisenberg picture. The quantum observables are represented by operators, and either state vectors evolve and operators remain still (Schrödinger), or operators evolve and state vectors remain still (Heisenberg). The distinction may feel a bit more subtle on the classical level, where the observables aren’t “reified”, so to speak, but it is still possible.
Measuring all necessary observables one determines the instantaneous state of the system. To predict the values of observables in a different instant, one needs to calculate the evolution of the instantaneous state, or equivalently to find out the permanent state.
Now there’s a problem already on the classical level: the time. We know that the microscopic laws are invariant with respect to the Lorentz transformation, which mix time and space, so it has no sense to treat time and space so differently (the former as a parameter of evolution and the latter as an observable), unless one is dealing with statistical physics where time is really special. Since the Hamiltonian formalism does treat space and time differently, the Lorentz invariance isn’t manifest there and the relativistic theories look awkward. So to do relativistic physics efficiently, either one leaves the Hamiltonian formulation, or turns from mechanics to field theory (where time and space are both parameters). However the Hamiltonian formulation is needed for the standard formulation of quantum theory. The move to field theory does help in the classical physics, but one has to resuscitate the crucial role of time at the moment of quantisation, and then the elegance and Lorentz invariance is lost again.
Another problem comes with general relativity. The general relativity is formulated in such a way that neither time nor spatial coordinates have any physical meaning: any coordinates can be used to address the spacetime points, and no set of coordinates is prefered by the laws of nature. This is called general covariance and has important consequences. Strictly speaking, there isn’t the time in general relativity. We can consider different times measured by particular clocks, but those are clearly not different from other observables.
Nevertheless, the Hamiltonian formalism can be salvaged. It’s done by adding the time (and its associated momentum, which may or may not be interpreted as energy) to the phase space. (In the field theory, one adds also the spatial coordinates, but I’ll limit myself to mechanics here.) The phase space has now two dimension more. The permanent (Heisenberg) states now correspond to trajectories q(τ), where the original time t is contained in q. The parameter τ has no physical meaning and the trajectory q(τ) can be reparametrised, while the state remains the same. For most realistic systems, one can choose such a parametrisation where t=τ, but there is no need to do so. This is the relativistic Hamiltonian formalism, whose field-theoretic version is used in attempts to quantise gravity (loop gravitists do that, string theorists do not).
The relativistic Hamiltonian formalism leads to surprising simplification of the Hamilton equations (at least when written in a coordinate-independent form) and Hamilton-Jacobi equations (written in any form). The Lorentz invariance is manifest in this formalism, too. Those facts suggest that this version of the formalism is closer to the real structure of nature than the standard, time-chauvinistic Hamiltonian formalism. An important point is that the notion of instantaneous state has no sense in the relativistic Hamiltonian formalism. Time and coordinates are treated equally, and to ask “in what state the system was at moment t” has roughly as much sense as to ask “in what state the system was at point x”.
(Notice that the usual talk about MWI is done using the Schrödinger picture. It looks a lot less intuitive and clear in a Heisenberg picture. To be fair, the collapse postulate in the Heisenberg picture is litterally bizarre.)
Forfeiting the right to parametrise evolution by time, one has to be sort of careful when asking questions. The question “what was the particle’s position x at time t” can be answered, but it’s no more a natural formulation of the question. The trajectories aren’t parametrised by t, they are parametrised by τ. (But to ask “what’s the position at τ” is even worse: τ is an unphysical, arbitrary, meaningless auxiliary parameter that should be elliminated from all questions of fact. Put so it may seem trivial, but untrained people tend to ask meaningless questions in general relativity precisely because they intuitively feel that the spacetime coordinates have some meaning, and it is often difficult to resolve the paradoxes they obtain from such questions.)
The natural form of a question is rather “what doublets x,t can be measured in the (permanent) state s?” But if x and t form a complete set of observables, one measurement of that doublet does determine the state s. Therefore, we can formulate an alternative question: “is it possible to measure both x1,t1 and x2,t2 on a single system?” In this formulation, the mention of state has been omitted. In practice, however, states are indexed by measurement outcomes and those two formulations are isomorphic. It may not be so in quantum theory.
In the standard Hamiltonian quantum theory (the one with time as parameter), one can measure only half of the observables compared to the classical theory—either the canonical coordinates, or the canonical momenta. Furthermore, there is no one-to-one correspondence between the state and the observable values. Nevertheless each observable has a probability distribution in any given instantaneous (Schrödinger) state. It’s possible to speak about Heisenberg states, but then, the probabilities which sum up to one are given by scalar products of the state vector and the eigenvectors of observable operators taken in one specific time instant. Measurement, as it happens, is supposed to be instantaneous. This poses a problem for relativistic theories, and consistent relativistic quantum mechanics is impossible (but see my remark at the bottom).
In particular, let’s ask what happens when two measurements are done. The orthodox interpretation says that during the first measurement the state collapses into the eigenstate of the measured observables, which corresponds to the observed values. We then ask for the probability of the second set of values, which can then be calculated from the new, collapsed wave function. The decoherence interpretations, and MWI in particular, tell us that (in the Schrödinger picture) during the measurement the observer’s own state vector becomes correlated. In the Heisenberg picture, this translates into a statement about the observable operators. The role of time can be obscured easily in such description, but in either interpretation, there have to be planes of simultaneous events defined in the space-time to normalise the state vector. Any such definition violates Lorentz invariance, of course. (See also the second remark.)
Like in the classical mechanics, one can resort to the relativistic Hamiltonian formalism. The formalism can be adopted to use in quantum theory, but now there are no observable operators q(t) with time-dependent eigenvectors: both q and t are (commuting) operators. There are indeed wave functions ψ(q,t), but their interpretation is not obvious. For details see here (the article partly overlaps with the one which I link in the remark 2, but gets deeper into the relativistic formalism). The space-time states discussed in the article are redundant—many distinct state vectors describe the same physical situation.
So what we have: either violation of the Lorentz symmetry, or a non-transparent representation of states. Of course, all physical questions in quantum physics can be formulated as questions of the second type as described four paragraphs above. One measures the observables twice (the first measurement is called preparation), and can then ask: “What’s the probability of measuring q2, when we have prepared the system into q1?” Which is equivalent to “what’s the probability of measuring q1 and q2 on the same system?”
And of course, there is the path integral formulation of quantum theory, which doesn’t even need to speak about state space, and is manifestly Lorentz-covariant. So it seems to me that the notion of a state of a system is redundand. The problem with collapse (which is really a problem—my original statement doesn’t mean an endorsement of collapse, although some readers may perceive it as such) doesn’t exist when we don’t speak about the states. Of course, the state vectors are useful in some calculations. I only don’t give them independent ontological status.
Remarks:
The fact that the quantum mechanics and relativity don’t fit together is often presented as a “feature, not bug”: it points out to the necessity of field theory, which, as we know, is a more precise description of the world. In my opinion, such declarations miss the mark, as they implicitly suggest that quantumness somehow doesn’t fit well with relativity and mechanics. But the problem here isn’t quantumness, the problem is the standard Hamiltonian formalism which singles out time as a special parameter. This can be concealed in the classical mechanics where, like time, dynamical variables are simple numbers, but it’s no longer true in quantum setting. Using the relativistic Hamiltonian formalism instead of the standard one, a Lorentz-invariant quantum mechanics can be consistently formulated.
In the decoherence interpretation, a measurement is thought of as an interaction between different parts of the world—the observer and the observed system—an interaction in principle no different from all other interactions. However, it is not so easy to describe such interaction. In any sensible definition the observer must retain memory of his observation. To do that, the interaction Hamiltonian has to be non-Hermitian or time-dependent; both are physically problematic properties. Non-Hermitian interactions are better choice, as they can model dissipation, which is actually the reason for memory in real observers. Another problem with measurement comes when one needs to think about resolution, as no detector can accurately measure the position of a particle with infinite precision. A finite precision of a position measurement is a trivial problem, but when it comes to time measurement, it can really be a mess. See this for a dicussion of a realistic measurement (collapse, but easily translatable into decoherence).
An outstanding summary. It reminded me of stuff I once knew and taught me one or two things I had missed until now. And in two parts to make it easy to upvote it twice.
But the purpose was to cast doubt on MWI. If you are merely pointing out that MWI is a non-relativistic theory, and hence cannot be exactly right, well, ok. But that just means we need a Lorentz invariant version of MWI. But I thought we already have one. Feynman’s sum-over-histories approach.
I guess my question is this: Are you just saying that MWI is wrong because it is not Lorentz invariant, or that it is wrong because it cannot be made Lorentz invariant, or that it is wrong because it cannot be made Lorentz invariant without giving up the interpretation that there are many worlds?
ETA: second question:
… there is the path integral formulation of quantum theory, which doesn’t even need to speak about state space …
I guess I don’t understand the path integral formulation then. I thought the paths being integrated were paths (trajectories) through a kind of state space. How am I wrong?
Are you just saying that MWI is wrong because it is not Lorentz invariant, or that it is wrong because it cannot be made Lorentz invariant, or that it is wrong because it cannot be made Lorentz invariant without giving up the interpretation that there are many worlds?
This is a difficult question. I have written the disclaimer above the grandparent precisely because I am not able to demonstrate that MWI is wrong. I believe MWI can be made Lorentz invariant and retain its interpretation, for the price of losing its intuitive appeal and making it awkward. One can postulate some kind of Lorentz invariant measurement procedure (like the one suggested in articles I’ve linked to) and do the interpretational stuff on the level of observer. In the Schrödinger picture it looks nice—in the Heisenberg picture not so.
My attack doesn’t aim to MWI specifically. I think the objective collapse is even a greater problem. Partly, to include MWI in the statement was part of my dirty tactic to make the statement more prominent, since belief in MWI is accepted here as one of the rationality tests (hell, there is even a sequence about it). But I suspect that the very dispute between collapse and many worlds is an artifact of asking about the behaviour of objective states of the system, and if it is possible to avoid speaking about states, the problem disappears. I want to explain away what MWI proponents want to explain. To further justify my inclusion of MWI specifically in the formulation of my supposedly irrational belief, I can add that, unlike the MWI proponents, there are (and were since the very beginning of the quantum theory) Copenhagenists who accept that the collapse is only a mathematical tool useful within our imperfect understanding of nature and it has no independent ontological status. This is a position with which I sympathise.
But I thought we already have one. Feynman’s sum-over-histories approach.
Could you explain in more detail?
I thought the paths being integrated were paths (trajectories) through a kind of state space.
When the path integral formulation is derived from the standard formulation, one integrates over paths in the phase space. However the integrations over momenta can be performed exactly and one is left with the integration over paths in the configuration space only (which is half of the phase space). This is the form which is prefered, as after the integration sign stands the exponential of action, which is a functional of the classical trajectory or field configuration (we can call both path). These paths needn’t solve the equations of motion, so there even isn’t a correspondence path—state.
We experience a classical world. To explain this “away” would be bad. The broadest interpretation of the phrase “many worlds” is that there are many classical worlds equally real to the single world we experience. Surely you accept this. There are questions of how real is this classical world and where it comes from. The decoherence program tries to address this, though I understand it to be incomplete, or at least controversial.
What gets worse when you move to QFT? You seem concerned with what is ontologically fundamental. The classical states are not ontologically fundamental in ordinary QM. If that’s what you mean by MWI...well, you already admitted to being a troll.
I’m not so concerned about fundamental ontology, so I’m happy to talk about QFT as a bunch of ordinary QM systems, one for each reference frame. The decomposition into classical states is not the same in each frame (ie, is not relativistically covariant). Is this a problem? Isn’t the situation of ordinary QM already almost this bad? In ordinary QM, you can give states classical names, but they don’t actually evolve classically. The macroscale classical worlds that do evolve classically are pretty fuzzy.
[T]here are many classical worlds equally real to the single world we experience. Surely you accept this.
Surely? I don’t even know what it means. Words “real” and “experience” are close neighbours in my vocabulary, real unexperienced world sounds a lot like an oxymoron, at least if not based on a really strong argument.
What gets worse when you move to QFT?
Nothing. I have tried to (incompletely of course) explain the relation between the conventional and relativistic Hamiltonian formalism in case of mechanics, where it is slightly more intuitive and simpler. If you address my first remark, you have disinterpreted it. I don’t say that move to QFT isn’t justified, but that one conventional argument used to support this move isn’t good.
The classical states are not ontologically fundamental in ordinary QM. If that’s what you mean by MWI...
It isn’t. By MWI I mean probably the same thing as anybody else. Nothing particularly related to classical states. I have discussed classical states in order to give some background to my intuitions. My statement was that probably the quantum states are a redundant concept.
I’m happy to talk about QFT as a bunch of ordinary QM systems, one for each reference frame.
I would understand that QFT is a bunch of QM systems, one for each spacetime point. I don’t understand what reference frames do with it. In any fixed reference frame QFT has infinite number of degrees of freedom. Maybe you speak about momentum representation? I am confused.
I suspect that the very dispute between collapse and many worlds is an artifact of asking about the behaviour of objective states of the system, and if it is possible to avoid speaking about states, the problem disappears. I want to explain away what MWI proponents want to explain.
Amen to that. Whenever we cease believing we are working with models and doing phenomenology, and start believing we are dealing with reality and doing ontology; at that point we have stopped doing science and entered the realm of metaphysics.
But I thought we already have one [relativistic MWI]. Feynman’s sum-over-histories approach.
Could you explain in more detail?
Be forewarned that my physics is at the “QM and QFT for Dummies” level. But I thought that a slogan of “one Everett world = one Feynman diagram” had some validity.
At least if you think of really big diagrams. (>5%)
I can add that, unlike the MWI proponents, there are (and were since the very beginning of the quantum theory) Copenhagenists who accept that the collapse is only a mathematical tool useful within our imperfect understanding of nature and it has no independent ontological status.
How do these people interpret interaction-free measurements? Specifically, let’s observe one of the possible outcomes of the Elitzur-Vaidman bomb-tester thought experiment, namely the one that identifies a working bomb without exploding it. To describe this experiment in Copenhagen terms, we could say that the interaction between the photon wave function and the bomb has, as a measurement, collapsed the photon wave function to the upper arm of the interferometer. Since we actually see this result in the detector, and obtain useful information about the bomb from it, I don’t see how we can deny that the collapse has been observed as an actual process while still insisting on Copenhagen. (But I’m sure there is a way to do it, if there are actual physicists who hold this position.)
This relates to the discussion where you’ve apparently participated, and I am not sure whether I can say more. I am quite content with the prediction of the theory, and don’t trust much the feeling of need of further verbal explanation here. If I were pressed to say something, I would say that probably the present formalism of quantum theory isn’t particularly well suited for human intuition. After all, I believe we will get better formalism in future, whatever it means.
The feeling that the collapse is needed somehow to mediate the bomb’s interaction with the detector falls to the same category with the belief that light must propagate in some medium, or a feeling that there must be some absolute time. Such intuitions are sometimes correct, more often wrong.
Based on my experience, most of the ordinary physicists don’t think interpretations of QM are a big issue. It isn’t discussed too often, people are content to do the calculations most of the time. Of course, this may be different among the first-rank researchers.
Just to clarify: in that discussion, I claimed that the bomb tester thought-experiment doesn’t pose any principal difficulty for Copenhagen relative to the standard variations on the double-slit experiment, so that might seem to contradict what I write here. What I meant to say there is that the main feature of the bomb-tester, namely the interaction-free measurement, is also featured in a less salient way in these classic though-experiments, so that Copenhagen also makes sense for the bomb tester if you accept that it makes sense at all.
But if I may ask, how would you reply to the following statement? “Consider the case when we have a dud bomb, and a case when we have a working bomb that doesn’t explode. There is an observable difference between what the detector shows in these outcomes, so replacing the dud bomb with a working one changed the system in a measurable way. We call this change—whatever exactly it might be—collapse.”
Do you believe that this statement would be flawed, or that it is, after all, somehow compatible with the idea that “the collapse is only a mathematical tool”?
Comparing a system with a dud to a system with a working bomb is comparing two different systems, or the same system in two instances with different initial conditions, and thus doesn’t relate to the collapse. I suppose you rather had in mind a statement: “Consider two experiments with a working bomb, and in one the bomb explodes, while in the second it doesn’t. There is an observable difference...”
Well, it is undeniable that there is a difference. The two systems were the same in the beginning and are different in the end. There are three conventional explanations. 1) The systems were different all way long, but in the beginning the difference was invisible for us (hidden parameters). 2) The difference emerged from a non-deterministic process before or during the measurement (collapse). 3) There is no difference, but we see only a portion of reality after the measurement, and a different one in each of the cases (many worlds).
I suggest fourth point of view: Don’t ask in what state the system is, this is meaningless. Ask only what measurement outcomes are possible, given the outcomes we had from the already performed measurements. If you do that, there is no paradox to solve.
There are three conventional explanations. 1) The systems were different all way long, but in the beginning the difference was invisible for us (hidden parameters). 2) The difference emerged from a non-deterministic process before or during the measurement (collapse). [...]
Actually, that’s the distinction I missed! The notion of “collapse” specifically refers to a non-deterministic process, not to a deterministic process that would at some point reveal the previously existing hidden variables.
I suggest fourth point of view: Don’t ask in what state the system is, this is meaningless. Ask only what measurement outcomes are possible, given the outcomes we had from the already performed measurements. If you do that, there is no paradox to solve.
That would basically be the “ensemble interpretation,” right? The theory tells you the probability distribution of outcomes, which you’ll see if you repeat the experiment prepared the same way a bunch of times (frequentism!), and that’s all there is to it. I do have a lot of sympathy for that view, as you might guess from the recent discussion of subjective probabilities, though I cannot say that my superficial understanding of QM gives me much confidence in any views I might hold about it.
The theory tells you the probability distribution of outcomes, which you’ll see if you repeat the experiment prepared the same way a bunch of times (frequentism!), and that’s all there is to it.
Well, the quantum probabilities are certainly frequentist. However, I don’t suppose strict Bayesians deny that there are probabilities with frequentist interpretation. I am also not sure about the label ensemble interpretation. It seems that its proponents somehow deny the validity of QM for small, non-ensemblish systems, which is a position I don’t subscribe to. After all, both collapse and many-world interpretations are no more Bayesian and no less frequentist than the ensemble one. The hidden parameters are deterministic, but have their own well known problems.
As for the frequentist-Bayes controversy, although I am probably more than you sympathetic to the Bayesian position, I have some sympathy for frequentism. I think both interpretation can coexist, with different sensible meanings of “probability”.
Disclaimer: If I had something well thought through, consistent, not vague and well supported, I would be sending it to Phys.Rev. instead of using it for karma-mining in the Irrationality thread on LW. Also, I don’t know your background in physics, so I will probably either unnecessarily spend some time explaining banalities, or leave something crucial unexplained, or both. And I am not sure how much of what I have written is relevant. But let me try.
The standard formulation of the quantum theory is based on the Hamiltonian formalism. In its classical variant, it relies on the phase space, which is coordinatised by dynamical variables (or observables; the latter term is more frequent in the quantum context). The observables are conventionally divided into pairs of canonical coordinates and momenta. The set of observables is called complete if their values determine the points in the phase space uniquely.
I will distinguish between two notions of state of a physical system. First, the instantaneous state corresponds to a point in the phase space. Such a state evolves, which means that as time passes, the point moves through the phase space along a trajectory. It has sense to say “the system at time t is in instantaneous state s” or “the instantaneous state s corresponds to the set of observables q”. In the quantum mechanics, the instantaneous state is described by state vectors in the Schrödinger picture.
Second, the permanent state is fixed and corresponds to a parametrised curve s=s(t). It has sense to say “the system in the state s corresponds to observable values q(t)”. In quantum mechanics, this is described by the state vectors in the Heisenberg picture. The quantum observables are represented by operators, and either state vectors evolve and operators remain still (Schrödinger), or operators evolve and state vectors remain still (Heisenberg). The distinction may feel a bit more subtle on the classical level, where the observables aren’t “reified”, so to speak, but it is still possible.
Measuring all necessary observables one determines the instantaneous state of the system. To predict the values of observables in a different instant, one needs to calculate the evolution of the instantaneous state, or equivalently to find out the permanent state.
Now there’s a problem already on the classical level: the time. We know that the microscopic laws are invariant with respect to the Lorentz transformation, which mix time and space, so it has no sense to treat time and space so differently (the former as a parameter of evolution and the latter as an observable), unless one is dealing with statistical physics where time is really special. Since the Hamiltonian formalism does treat space and time differently, the Lorentz invariance isn’t manifest there and the relativistic theories look awkward. So to do relativistic physics efficiently, either one leaves the Hamiltonian formulation, or turns from mechanics to field theory (where time and space are both parameters). However the Hamiltonian formulation is needed for the standard formulation of quantum theory. The move to field theory does help in the classical physics, but one has to resuscitate the crucial role of time at the moment of quantisation, and then the elegance and Lorentz invariance is lost again.
Another problem comes with general relativity. The general relativity is formulated in such a way that neither time nor spatial coordinates have any physical meaning: any coordinates can be used to address the spacetime points, and no set of coordinates is prefered by the laws of nature. This is called general covariance and has important consequences. Strictly speaking, there isn’t the time in general relativity. We can consider different times measured by particular clocks, but those are clearly not different from other observables.
Nevertheless, the Hamiltonian formalism can be salvaged. It’s done by adding the time (and its associated momentum, which may or may not be interpreted as energy) to the phase space. (In the field theory, one adds also the spatial coordinates, but I’ll limit myself to mechanics here.) The phase space has now two dimension more. The permanent (Heisenberg) states now correspond to trajectories q(τ), where the original time t is contained in q. The parameter τ has no physical meaning and the trajectory q(τ) can be reparametrised, while the state remains the same. For most realistic systems, one can choose such a parametrisation where t=τ, but there is no need to do so. This is the relativistic Hamiltonian formalism, whose field-theoretic version is used in attempts to quantise gravity (loop gravitists do that, string theorists do not).
The relativistic Hamiltonian formalism leads to surprising simplification of the Hamilton equations (at least when written in a coordinate-independent form) and Hamilton-Jacobi equations (written in any form). The Lorentz invariance is manifest in this formalism, too. Those facts suggest that this version of the formalism is closer to the real structure of nature than the standard, time-chauvinistic Hamiltonian formalism. An important point is that the notion of instantaneous state has no sense in the relativistic Hamiltonian formalism. Time and coordinates are treated equally, and to ask “in what state the system was at moment t” has roughly as much sense as to ask “in what state the system was at point x”.
(Notice that the usual talk about MWI is done using the Schrödinger picture. It looks a lot less intuitive and clear in a Heisenberg picture. To be fair, the collapse postulate in the Heisenberg picture is litterally bizarre.)
Forfeiting the right to parametrise evolution by time, one has to be sort of careful when asking questions. The question “what was the particle’s position x at time t” can be answered, but it’s no more a natural formulation of the question. The trajectories aren’t parametrised by t, they are parametrised by τ. (But to ask “what’s the position at τ” is even worse: τ is an unphysical, arbitrary, meaningless auxiliary parameter that should be elliminated from all questions of fact. Put so it may seem trivial, but untrained people tend to ask meaningless questions in general relativity precisely because they intuitively feel that the spacetime coordinates have some meaning, and it is often difficult to resolve the paradoxes they obtain from such questions.)
The natural form of a question is rather “what doublets x,t can be measured in the (permanent) state s?” But if x and t form a complete set of observables, one measurement of that doublet does determine the state s. Therefore, we can formulate an alternative question: “is it possible to measure both x1,t1 and x2,t2 on a single system?” In this formulation, the mention of state has been omitted. In practice, however, states are indexed by measurement outcomes and those two formulations are isomorphic. It may not be so in quantum theory.
In the standard Hamiltonian quantum theory (the one with time as parameter), one can measure only half of the observables compared to the classical theory—either the canonical coordinates, or the canonical momenta. Furthermore, there is no one-to-one correspondence between the state and the observable values. Nevertheless each observable has a probability distribution in any given instantaneous (Schrödinger) state. It’s possible to speak about Heisenberg states, but then, the probabilities which sum up to one are given by scalar products of the state vector and the eigenvectors of observable operators taken in one specific time instant. Measurement, as it happens, is supposed to be instantaneous. This poses a problem for relativistic theories, and consistent relativistic quantum mechanics is impossible (but see my remark at the bottom).
In particular, let’s ask what happens when two measurements are done. The orthodox interpretation says that during the first measurement the state collapses into the eigenstate of the measured observables, which corresponds to the observed values. We then ask for the probability of the second set of values, which can then be calculated from the new, collapsed wave function. The decoherence interpretations, and MWI in particular, tell us that (in the Schrödinger picture) during the measurement the observer’s own state vector becomes correlated. In the Heisenberg picture, this translates into a statement about the observable operators. The role of time can be obscured easily in such description, but in either interpretation, there have to be planes of simultaneous events defined in the space-time to normalise the state vector. Any such definition violates Lorentz invariance, of course. (See also the second remark.)
(Comment too long, continued in a subcomment.)
Like in the classical mechanics, one can resort to the relativistic Hamiltonian formalism. The formalism can be adopted to use in quantum theory, but now there are no observable operators q(t) with time-dependent eigenvectors: both q and t are (commuting) operators. There are indeed wave functions ψ(q,t), but their interpretation is not obvious. For details see here (the article partly overlaps with the one which I link in the remark 2, but gets deeper into the relativistic formalism). The space-time states discussed in the article are redundant—many distinct state vectors describe the same physical situation.
So what we have: either violation of the Lorentz symmetry, or a non-transparent representation of states. Of course, all physical questions in quantum physics can be formulated as questions of the second type as described four paragraphs above. One measures the observables twice (the first measurement is called preparation), and can then ask: “What’s the probability of measuring q2, when we have prepared the system into q1?” Which is equivalent to “what’s the probability of measuring q1 and q2 on the same system?”
And of course, there is the path integral formulation of quantum theory, which doesn’t even need to speak about state space, and is manifestly Lorentz-covariant. So it seems to me that the notion of a state of a system is redundand. The problem with collapse (which is really a problem—my original statement doesn’t mean an endorsement of collapse, although some readers may perceive it as such) doesn’t exist when we don’t speak about the states. Of course, the state vectors are useful in some calculations. I only don’t give them independent ontological status.
Remarks:
The fact that the quantum mechanics and relativity don’t fit together is often presented as a “feature, not bug”: it points out to the necessity of field theory, which, as we know, is a more precise description of the world. In my opinion, such declarations miss the mark, as they implicitly suggest that quantumness somehow doesn’t fit well with relativity and mechanics. But the problem here isn’t quantumness, the problem is the standard Hamiltonian formalism which singles out time as a special parameter. This can be concealed in the classical mechanics where, like time, dynamical variables are simple numbers, but it’s no longer true in quantum setting. Using the relativistic Hamiltonian formalism instead of the standard one, a Lorentz-invariant quantum mechanics can be consistently formulated.
In the decoherence interpretation, a measurement is thought of as an interaction between different parts of the world—the observer and the observed system—an interaction in principle no different from all other interactions. However, it is not so easy to describe such interaction. In any sensible definition the observer must retain memory of his observation. To do that, the interaction Hamiltonian has to be non-Hermitian or time-dependent; both are physically problematic properties. Non-Hermitian interactions are better choice, as they can model dissipation, which is actually the reason for memory in real observers. Another problem with measurement comes when one needs to think about resolution, as no detector can accurately measure the position of a particle with infinite precision. A finite precision of a position measurement is a trivial problem, but when it comes to time measurement, it can really be a mess. See this for a dicussion of a realistic measurement (collapse, but easily translatable into decoherence).
An outstanding summary. It reminded me of stuff I once knew and taught me one or two things I had missed until now. And in two parts to make it easy to upvote it twice.
But the purpose was to cast doubt on MWI. If you are merely pointing out that MWI is a non-relativistic theory, and hence cannot be exactly right, well, ok. But that just means we need a Lorentz invariant version of MWI. But I thought we already have one. Feynman’s sum-over-histories approach.
I guess my question is this: Are you just saying that MWI is wrong because it is not Lorentz invariant, or that it is wrong because it cannot be made Lorentz invariant, or that it is wrong because it cannot be made Lorentz invariant without giving up the interpretation that there are many worlds?
ETA: second question:
I guess I don’t understand the path integral formulation then. I thought the paths being integrated were paths (trajectories) through a kind of state space. How am I wrong?
This is a difficult question. I have written the disclaimer above the grandparent precisely because I am not able to demonstrate that MWI is wrong. I believe MWI can be made Lorentz invariant and retain its interpretation, for the price of losing its intuitive appeal and making it awkward. One can postulate some kind of Lorentz invariant measurement procedure (like the one suggested in articles I’ve linked to) and do the interpretational stuff on the level of observer. In the Schrödinger picture it looks nice—in the Heisenberg picture not so.
My attack doesn’t aim to MWI specifically. I think the objective collapse is even a greater problem. Partly, to include MWI in the statement was part of my dirty tactic to make the statement more prominent, since belief in MWI is accepted here as one of the rationality tests (hell, there is even a sequence about it). But I suspect that the very dispute between collapse and many worlds is an artifact of asking about the behaviour of objective states of the system, and if it is possible to avoid speaking about states, the problem disappears. I want to explain away what MWI proponents want to explain. To further justify my inclusion of MWI specifically in the formulation of my supposedly irrational belief, I can add that, unlike the MWI proponents, there are (and were since the very beginning of the quantum theory) Copenhagenists who accept that the collapse is only a mathematical tool useful within our imperfect understanding of nature and it has no independent ontological status. This is a position with which I sympathise.
Could you explain in more detail?
When the path integral formulation is derived from the standard formulation, one integrates over paths in the phase space. However the integrations over momenta can be performed exactly and one is left with the integration over paths in the configuration space only (which is half of the phase space). This is the form which is prefered, as after the integration sign stands the exponential of action, which is a functional of the classical trajectory or field configuration (we can call both path). These paths needn’t solve the equations of motion, so there even isn’t a correspondence path—state.
We experience a classical world. To explain this “away” would be bad. The broadest interpretation of the phrase “many worlds” is that there are many classical worlds equally real to the single world we experience. Surely you accept this. There are questions of how real is this classical world and where it comes from. The decoherence program tries to address this, though I understand it to be incomplete, or at least controversial.
What gets worse when you move to QFT? You seem concerned with what is ontologically fundamental. The classical states are not ontologically fundamental in ordinary QM. If that’s what you mean by MWI...well, you already admitted to being a troll.
I’m not so concerned about fundamental ontology, so I’m happy to talk about QFT as a bunch of ordinary QM systems, one for each reference frame. The decomposition into classical states is not the same in each frame (ie, is not relativistically covariant). Is this a problem? Isn’t the situation of ordinary QM already almost this bad? In ordinary QM, you can give states classical names, but they don’t actually evolve classically. The macroscale classical worlds that do evolve classically are pretty fuzzy.
Surely? I don’t even know what it means. Words “real” and “experience” are close neighbours in my vocabulary, real unexperienced world sounds a lot like an oxymoron, at least if not based on a really strong argument.
Nothing. I have tried to (incompletely of course) explain the relation between the conventional and relativistic Hamiltonian formalism in case of mechanics, where it is slightly more intuitive and simpler. If you address my first remark, you have disinterpreted it. I don’t say that move to QFT isn’t justified, but that one conventional argument used to support this move isn’t good.
It isn’t. By MWI I mean probably the same thing as anybody else. Nothing particularly related to classical states. I have discussed classical states in order to give some background to my intuitions. My statement was that probably the quantum states are a redundant concept.
I would understand that QFT is a bunch of QM systems, one for each spacetime point. I don’t understand what reference frames do with it. In any fixed reference frame QFT has infinite number of degrees of freedom. Maybe you speak about momentum representation? I am confused.
Amen to that. Whenever we cease believing we are working with models and doing phenomenology, and start believing we are dealing with reality and doing ontology; at that point we have stopped doing science and entered the realm of metaphysics.
Be forewarned that my physics is at the “QM and QFT for Dummies” level. But I thought that a slogan of “one Everett world = one Feynman diagram” had some validity. At least if you think of really big diagrams. (>5%)
prase:
How do these people interpret interaction-free measurements? Specifically, let’s observe one of the possible outcomes of the Elitzur-Vaidman bomb-tester thought experiment, namely the one that identifies a working bomb without exploding it. To describe this experiment in Copenhagen terms, we could say that the interaction between the photon wave function and the bomb has, as a measurement, collapsed the photon wave function to the upper arm of the interferometer. Since we actually see this result in the detector, and obtain useful information about the bomb from it, I don’t see how we can deny that the collapse has been observed as an actual process while still insisting on Copenhagen. (But I’m sure there is a way to do it, if there are actual physicists who hold this position.)
This relates to the discussion where you’ve apparently participated, and I am not sure whether I can say more. I am quite content with the prediction of the theory, and don’t trust much the feeling of need of further verbal explanation here. If I were pressed to say something, I would say that probably the present formalism of quantum theory isn’t particularly well suited for human intuition. After all, I believe we will get better formalism in future, whatever it means.
The feeling that the collapse is needed somehow to mediate the bomb’s interaction with the detector falls to the same category with the belief that light must propagate in some medium, or a feeling that there must be some absolute time. Such intuitions are sometimes correct, more often wrong.
Based on my experience, most of the ordinary physicists don’t think interpretations of QM are a big issue. It isn’t discussed too often, people are content to do the calculations most of the time. Of course, this may be different among the first-rank researchers.
Just to clarify: in that discussion, I claimed that the bomb tester thought-experiment doesn’t pose any principal difficulty for Copenhagen relative to the standard variations on the double-slit experiment, so that might seem to contradict what I write here. What I meant to say there is that the main feature of the bomb-tester, namely the interaction-free measurement, is also featured in a less salient way in these classic though-experiments, so that Copenhagen also makes sense for the bomb tester if you accept that it makes sense at all.
But if I may ask, how would you reply to the following statement? “Consider the case when we have a dud bomb, and a case when we have a working bomb that doesn’t explode. There is an observable difference between what the detector shows in these outcomes, so replacing the dud bomb with a working one changed the system in a measurable way. We call this change—whatever exactly it might be—collapse.”
Do you believe that this statement would be flawed, or that it is, after all, somehow compatible with the idea that “the collapse is only a mathematical tool”?
Comparing a system with a dud to a system with a working bomb is comparing two different systems, or the same system in two instances with different initial conditions, and thus doesn’t relate to the collapse. I suppose you rather had in mind a statement: “Consider two experiments with a working bomb, and in one the bomb explodes, while in the second it doesn’t. There is an observable difference...”
Well, it is undeniable that there is a difference. The two systems were the same in the beginning and are different in the end. There are three conventional explanations. 1) The systems were different all way long, but in the beginning the difference was invisible for us (hidden parameters). 2) The difference emerged from a non-deterministic process before or during the measurement (collapse). 3) There is no difference, but we see only a portion of reality after the measurement, and a different one in each of the cases (many worlds).
I suggest fourth point of view: Don’t ask in what state the system is, this is meaningless. Ask only what measurement outcomes are possible, given the outcomes we had from the already performed measurements. If you do that, there is no paradox to solve.
prase:
Actually, that’s the distinction I missed! The notion of “collapse” specifically refers to a non-deterministic process, not to a deterministic process that would at some point reveal the previously existing hidden variables.
That would basically be the “ensemble interpretation,” right? The theory tells you the probability distribution of outcomes, which you’ll see if you repeat the experiment prepared the same way a bunch of times (frequentism!), and that’s all there is to it. I do have a lot of sympathy for that view, as you might guess from the recent discussion of subjective probabilities, though I cannot say that my superficial understanding of QM gives me much confidence in any views I might hold about it.
Well, the quantum probabilities are certainly frequentist. However, I don’t suppose strict Bayesians deny that there are probabilities with frequentist interpretation. I am also not sure about the label ensemble interpretation. It seems that its proponents somehow deny the validity of QM for small, non-ensemblish systems, which is a position I don’t subscribe to. After all, both collapse and many-world interpretations are no more Bayesian and no less frequentist than the ensemble one. The hidden parameters are deterministic, but have their own well known problems.
As for the frequentist-Bayes controversy, although I am probably more than you sympathetic to the Bayesian position, I have some sympathy for frequentism. I think both interpretation can coexist, with different sensible meanings of “probability”.