Questions about Empirical Equivalence in STR and QT

I’m going to start off by saying that I may or may not have a correct understanding of the terms I’m using. Please bear that in mind and correct me if the way I’m expressing the terms is incorrect.

With that said, if I’m not mistaken, in Lorentzian vs Einsteinian versions of STR both the postulates and the mechanisms, i.e., equations, are different. But with QT it’s not so clear to me. As I understand it, in QT there seems to be a similar situation as in STR when comparing Pilot Wave and Copenhagen interpretations in that both the postulates and mechanisms differ.

But what’s not real clear to me is whether that’s the case for the postulates and mechanisms with other interpretations of QT, or whether it’s just the postulates that differ in the other cases? In other words, what I’m having difficulty figuring out is how many empirically equivalent mechanisms, i.e., different equations in the overall formulation with the same predictions, there are in QT besides Bohmian mechanics and textbook Quantum mechanics.

If anyone who can wants to help me out here I’d be grateful. No need to get into great detail. Just a general idea would be great of which ones of the different interpretations there are that have different equations than the orthodox textbook quantum mechanics.

Hello again Jim! Can’t say I entirely understand what you’re asking here, but I’ll give it a shot.

But a couple things jump out at me right off the bat:

This (conflating “mechanisms” and “equations”) seems entirely wrong to me. The equations of physics describe the behaviours of physical systems, or relationships between quantities characterizing those systems. This is conceptually different from mechanisms, which to me means the causal influences that effect those behaviours or relationships. They’re related, but two different theories or interpretations can clearly have the same equations but different mechanisms (different interpretations of QM being an example).

Arguably the postulates of orthodox STR vs a Lorentzian interpretation amount to exactly the same claim, which is that the laws of physics are Lorentz invariant. Granted, it’s more usually said that the postulates of orthodox STR are (1) the laws of physics are the same in all inertial frames and (2) the speed of light is constant; and that the Lorentzian interpretation rejects either or both of those postulates by privileging a certain inertial frame - but that relies on assuming an “ontologically heavy” view of what constitutes an inertial frame, for example. If laws of physics, inertial frames, and the speed of light in an inertial frame are understood in terms of empirical quantities only, then those two postulates are equally true in the Lorentzian interpretation.

Going further, orthodox STR and the Lorentzian interpretation have exactly the same equations, and as far as I’m aware this isn’t even contested. Under the Lorentzian interpretation those equations only describe what is going on in an “ontologically heavy” sense in one privileged reference frame, but they remain empirically valid in every inertial frame, just as in orthodox STR.

As far as mechanisms go, again arguably, they can be said to be the same between orthodox and Lorentzian STR (though there’s a hitch). In Lorentzian STR, for example, when an object speeds up relative to the privileged frame, it’s just a consequence of the Lorentz-invariant dynamics governing the interactions between the particles that make up the object and the forces holding them together that the object length-contracts in the direction of its motion, and its internal processes slow down. (Look up John Bell’s “How to Teach Special Relativity” for the details.) This mechanism is just as valid in orthodox STR, only then it applies in any inertial frame (and you relativise causal explanations to frames); this seems to be Einstein’s view.

The hitch is that in orthodox STR, some people seem to not regard the dynamical interactions between particles and forces as the real mechanisms behind relativistic effects - instead, the real mechanism is the Minkowskian geometry of space-time. (In my own opinion, this doesn’t really make sense, nor does it make sense to relativise causal explanations to frames, which is part of why I’m a Lorentzian.)

So between orthodox and Lorentzian STR, the postulates and equations are the same. The mechanisms maybe are different, but don’t have to be (except that the mechanisms hold in reality only in the privileged reference frame in Lorentzian STR, and just in appearance in the other frames). All inertial frames are entirely empirically equivalent in both interpretations, and the only big difference is whether all inertial frames are ontologically equivalent. Orthodox STR says yes, Lorentzian says no.

I’ll get back to you on QM another time (probably tomorrow).

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This is closer to the truth. What follows is a long explanation in which I basically end up agreeing with you, but I just want to make sure everyone is clear on the details…

Part of the difficulty in quantum mechanics is that “orthodox” or “textbook” QM is highly ambiguous about its ontological commitments. In practice, your everyday physicist sometimes treats the wavefunction / quantum state as an objective physical reality itself, and sometimes treats it merely as a representation of our knowledge about objective physical reality. And they routinely shift back and forth between these views depending on the context. (I recommend David Wallace’s paper What is orthodox quantum mechanics? to understand the situation there better.)

Because of this ambiguity, we can find views ranging from the consciousness-causes-collapse interpretation to the many-worlds interpretation all being claimed as “just quantum mechanics, nothing added”, and a lot of confusion ensues. Part of my point here is to reinforce my earlier assertion that entirely different explanations and mechanisms can be associated to the same equations by different interpretations: orthodox QM and the consciousness-causes-collapse interpretation both just have the Schrodinger equation and the Born rule as their equations, but posit different mechanisms. But to return to your question…

The two different ways that orthodox QM handles the quantum state correspond to what you might consider its two postulates, broadly speaking: (1) a system has a quantum state evolving according to the Schrodinger equation, and (2) we can measure observable properties of the system, and the probability of getting a given result is calculated by the Born rule using the appropriate operator, which is associated with that observable property, on the quantum state. So while it doesn’t come out and say it, orthodox QM implicitly assumes that our macroscopic world, including observable physical properties, emerges in some unspecified way from the quantum state, and it takes as axiomatic that observables correspond to certain operators on the quantum state.

Now compare this to Bohmian mechanics, which we can also say has two postulates: (1) a system (actually the whole universe) has a quantum state that evolves according to the Schrodinger equation, and (2’) that system (again, actually the universe) is made of particles, and those particles move according to the guidance equation. (Here the facts that there’s a correspondence between observable physical properties and operators on the quantum state, and that we can use the Born rule to predict the results of measurements, are consequences of the theory rather than fundamental axioms.)

So orthodox QM and Bohmian mechanics share postulate (1) and the corresponding Schrodinger equation, while they differ in that orthodox QM has postulate (2) and the Born rule, while Bohmian mechanics has postulate (2’) and the guidance equation. (To be more precise, orthodox QM has (2) and the Born rule as fundamental, while Bohmian mechanics has them as a consequence of its postulates.) And of course, Bohmian mechanics has a mechanism (the guiding of the particles by the quantum state, as described by the guidance equation) which orthodox QM does not.

In terms of equations:

  • Orthodox QM has the Schrodinger equation and the Born rule.
  • Bohmian mechanics has the Schrodinger equation and the guidance equation (and other pilot-wave theories are similar, with their own versions of the guidance equation),
  • Objective collapse theories have a modification of the Schrodinger equation to include the collapse process, and typically also have an equation that relates the quantum state to something that makes up the universe (e.g. a mass density or charge density) the way particles make up the Bohmian universe. But it should be noted that objective collapse theories are not empirically equivalent to orthodox QM, strictly speaking - though much of the region where they make different predictions from orthodox QM remain unexplored (i.e. they haven’t been ruled out yet, though see here).
  • The many-worlds interpretation as usually described has only the Schrodinger equation.
  • Other versions of the many-worlds interpretation can add an equation relating the quantum state to the physical constituents of the universe, as in objective collapse theories (and I would argue that it should add such an equation, or else it faces certain difficulties making it highly implausible if not incoherent - but that’s a different story).
  • You could even have a version of many-worlds that gets rid of the Schrodinger equation in favour of equations governing the dynamics of individual worlds and interactions between the parallel worlds, though the details aren’t all worked out.
  • Most other interpretations either can be shoehorned into one of the above categories (pilot-wave, objective collapse, or many-worlds), or uses the same equations as orthodox QM, just giving them a different interpretation.

It’s much more difficult to say anything about how the proposed postulates or mechanisms of the theories/interpretations differ without going into the details of every theory. Hope that response is something in the vicinity of what you were looking for.


First I want to say thanks for taking the time to help me to get a better understand of these things. I think I get the QT post, and as far as I can tell that’s what I was looking for. The post about STR I’m not sure I completely follow. But before I get to that I have some questions about the way terms are being used.

When it comes to STR, GTR, and QT it seems to me there are generally two activities going on. One is describing through mathematics observable effects of the interactions of matter and energy in space. The other is an attempt to answer the question of what the underlying unobservable reality is that pertains to those observed effects.

If that’s the case I just need some clarification on which specific terms apply to each of those two, what I would consider, separate activities. If that’s not the case maybe you could help me get a better idea of what’s going on. I’m also not sure if there is a distinction between formula and equation. I’m assuming they mean pretty much the same thing in this context and can be used interchangeably. But maybe that’s not the case?

I would say I agree here in the sense that I consider (1) describing what we observe and (2) explaining those observations via some theory as two separate activities. But it doesn’t have to be the case that the mathematics are strictly involved in (1) and that (2) is entirely about what is unobservable. For example, sometimes the most convenient way to mathematically describe some observations is to unify them with a theory that explains them, and describe that theory mathematically instead.

Furthermore, the boundary between observable and unobservable isn’t a clearly defined one - we can’t see or touch space, for example, but by moving through it and even just through seeing things in it, arguably we do observe space. When we observe gravitational lensing what we literally see is just images of stars / galaxies coming from different directions depending on whether the path of light passes close by the sun, for example. (Or see here for an even more dramatic example.) So do we literally observe gravitational lensing, or is lensing just part of a theory that explains what we do observe? In this way, while (1) and (2) are conceptually distinct, they are very tightly entangled - the very way that we describe our observations inevitably ends up relying on the theory we use to explain them.

It gets a little more complicated by the fact that most physicists aren’t concerned about being metaphysically precise with their theories (which is fine, at least as far as the practice of physics goes). A physicist is perfectly happy with the statement “energy curves spacetime” so long as he/she can relate that statement to what we can observe, without going into details like whether spacetime is a substance or not, or what exactly is the nature of the energy that curves spacetime (particles? fields? quantum state vector in an abstract Hilbert space of infinite dimension?), etc - leaving people like you or I with unanswered questions.

Loosely speaking, probably no distinction. Speaking more strictly, one might say a formula can refer to any mathematical expression (e.g. “a + b”) while an equation would only refer to one that expresses an equality between two different sub-expressions (e.g. “a + b = 0”).

Are you saying there is a conjunction of sorts between the two activities? As I see it, if (1) and (2) are separate and distinct activities either (1) is happening, or (2). I don’t see how it can be a bit of both. (1) may facilitate or inform (2), or visa versa. But if they are distinct activities to conjoin them in any sense to me is akin to saying something like “she’s a little bit pregnant.” WDYT?

In the statement you quoted, that is not what I said. What I said was that we can use mathematical statements in doing activity (2) (i.e. explaining what we observe via some theory); math isn’t confined to activity (1) (i.e. describing what we observe).

I wholeheartedly _dis_agree with this. As I indicated, sometimes - in fact very often - we describe what we observe in terms of the theory we use to explain it; i.e we do (1) and (2) at the same time. The fact that they are conceptually distinct activities does not preclude doing them simultaneously. (Walking and talking are distinct, yet I can do them at the same time.)

Let me see if I understand correctly. You’re saying that because in instances of doing (1) where information derived from doing (2) is utilized that we’re doing both (1) and (2)?

I’m saying that sometimes the way we describe our observations depends strongly on the theory we use to explain them - in some cases, we might not even be able to adequately describe the observations separately from the theory. In fact, ordinary everyday observation is like this - we don’t say “I see a pattern of light forming an image of a tree on my retinas”, we just say “I see a tree”. Similarly, when we observe a gravitationally lensed quasar, we typically skip over “our telescope recorded a pattern of four points of light being recieved from the sky in a cross shape around another pattern of light” to “we observed four images of the same distant quasar appearing around a galaxy in the foreground”. Yet recognizing that the four images were really the same object, and even that the spot of light in the middle of the images is a galaxy instead of something else, depends on our astronomical theories.

But, @Jim, before we get too far into this - do we need another thread devoted to the distinction between observation and inference and the way they are related in practice? I feel we’ve been over this ground before. I’d rather get back to relativity and quantum mechanics. :slight_smile:

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OK. I’m fine with setting that particular discussion aside for now. Thanks to you I think I now realize in STR that it’s not so much the difference in equations, but how those equations are applied that is different and that with STR empirical equivalence is mainly referring to the difference in ontology. Is that correct?

And would it be correct to say that Einstein’s formulation along with the help of Minkowskian spacetime reduces the steps needed in Lorentz’s formulation to get to the solution even though the equations are essentially the same?

Between orthodox vs. Lorentzian interpretations of STR, I would not even say there is a difference between the way the equations are applied per se, if by applied you mean the way they are used in practice. It’s really just how the equations are interpreted; i.e., what they are thought to mean at the level of ontology.

Only if a Lorentzian insisted on taking the extra steps of translating all equations back to the privileged reference frame, which really isn’t necessary. (In fact, since we don’t know our state of motion with respect to the privileged frame, it isn’t even possible!) The Lorentzian can easily consider the full 4D Minkowski spacetime as an abstraction from the real changing 3D universe, and work with it in the same way as would a physicist who takes the orthodox interpretation.

OK. But just to clarify, if comparing the original approach of Lorentzian SRT with that of the original Einsteinian SRT and the first formulations of Einsteinian SRT with the spacetime component added to it would it be correct to say that there were more steps involved in the way Lorentz approached it than the others? I’m asking because what I’ve read seems to indicate that such was the case.

This seems to be more of a question of how the theory was developed - am I right in that? - whereas my earlier comments pertained to how one would go about using the theory. And the answer to that is, I don’t know. What have you read indicating that was the case?

Indeed, I don’t know whether Lorentz himself actually held the kind of interpretation that I’ve been thinking of (which I should probably be calling neo-Lorentzian to be more precise) or if he instead held to some kind of mechanical aether theory.

I guess I got that impression from the last paragraph in Bell’s chapter on how to teach SR. Especially where he says, “This permits a very concise and elegant formulation of the theory, …”

The difference of style is that instead of inferring the experience of moving
observers from known and conjectured laws of physics, Einstein starts from
the hypothesis that the laws will look the same to all observers in uniform
motion. This permits a very concise and elegant formulation of the theory,
as often happens when one big assumption can be made to cover several less
big ones. There is no intention here to make any reservation whatever
about the power and precision of Einstein’s approach. But in my opinion
there is also something to be said for taking students along the road made
by Fitzgerald, Larmor, Lorentz and Poincare. The longer road sometimes
gives more familiarity with the country

I think Bell is speaking more about the pedagogical benefits of the approach he lays out in that paper, which is not about the Lorentzian interpretation per se. The “longer road” he takes is first to directly demonstrate the effects of length contraction and time dilation from the equations of electrodynamics in one reference frame in a fairly concrete way, then argue from those effects to the experiences of moving observers (showing that the laws of physics under consideration appear the same in the moving reference frame, so that we couldn’t tell if we are in the original rest frame or the moving frame), and then arrive at orthodox relativity by introducing the idea that no reference frame is ontologically privileged.

Of course you can also teach the Lorentzian interpretation in the same way (omitting the last step), but you don’t need to - and the Lorentzian interpretation also permits a concise and elegant formulation of the theory, in fact they very same one as the orthodox theory: it can be stated as “the laws of physics are Lorentz invariant.”

Oh, so he’s referring to the postulates when he says, " a very concise and elegant formulation?"

Yes, I think that is clear from the sentence immediately prior.

OK. I guess it’s hard to differentiate between when someone talks about conciseness, which I would assume is akin to simplicity, and elegance of the formulation of a theory if they’re referring to just the postulates or the equations. I’m assuming there are cases where it would apply to both?

Another thing that’s puzzling to me about the equations is concerning spacetime. Would there be a difference between equations of SR that were formulated using fused together Minkowskian spacetime and those that were formulated using unfused space and time? And would that be just those specific portions regarding space, time, and spacetime, or would it somehow cause some of the rest of the equations to be formulated differently?

Really not sure what you mean here. Concise just means expressing something in few words. So, sure, “akin to simplicity”, but not necessarily in the same way that simplicity is used of scientific theories (which would more commonly refer to something like postulating the fewest entities/interactions, or fewest kinds of entities/interactions, etc.).

There’s no difference in the equations. The difference, again, is in how those equations are interpreted.

Still not clear to me what’s going on. It seems to me that before Minkowski spacetime was introduced to Einstein’s or Lorentz’s SR it doesn’t seem like the Minkowski metric would have played a role in either of the earlier versions of SR. Would that not suggest a difference of some kind between earlier versions of SR without Minkowski spacetime and later versions with?