The Paradigm of Kinematics and Dynamics Must Yield to Causal Structure by Robert W. Spekkens

Hi again. Quick(er) replies on the other points...

* Ptolemy/Copernicus. The idea that these models only differ as to choice of reference frame is a very contemporary perspective. Certainly that is not how Copernicus saw what he was proposing. He instead thought he was proposing a distinct physical explanation for observable phenomena, and I think he was right to think that. But I feel like this thread is taking us farther away from the main issue you raised. I only raised it as a simple talking point contra operationalism, and it's become clear we're not as far apart there as I initially thought.

* The example of GRW vs (modified) MWI reminds me of the discussion about the "free will theorem" between Conway/Kochen and Tumulka/etc. I think that, if these extra degrees of freedom being posited in the modified MWI theory are considered beables, then the two theories posit different ontologies and (hence) should really be understood as physically distinct theories. Or if instead the extra degrees of freedom are not beables, then evidently this would be merely a different mathematical way to express the GRW evolution law. I think you want to say, on the contrary, that even if the extra degrees of freedom are beables, since they remain in some sense "hidden", the theories should be considered as members of the same equivalence class. Maybe. It's an interesting suggestion. My worry would be, though, that as in the "free will theorem" case, certain other properties (e.g., locality) could actually differ. But I should think about all this more.

* It seems strange to me to use the word "locality" to denote something that need not relate to spatially separated systems. No point arguing about terminology, though. Your different usage is noted.

* You've basically got what I was trying to suggest vis a vis the formulation of "kinematical locality" in some sense already presupposing it. It's not, though, exactly that I "want to say that systems are not primitive". I'm rather just trying to point out that, with the usual sort of formulation of kinematical locality, you *presume* the existence of identifiable spatially separated systems. But if you aren't already neck deep in knowing about QM, and somebody just comes up to you on the street and hands you a random two-particle wave function and asks you to figure out what kind of physical system it might describe, I don't think you'd ever come up with "well, obviously it's describing two particles, each of which exists in 3-space, but there are certain properties of the 2-particle system that don't supervene on the properties of the individuals." You'd just never find that there unless somehow you already knew that's what you were supposed to find. Thus, with standard formulations of "kinematical locality", I think it's a gross understatement (if not just an outright lie) to say that one of the weird things about QM is that it violates kinematical locality. No, what's weird is that the wave function, if a beable, is a nonlocal beable -- and worse, if the wf is all you've got in the theory, you have no local beables! But no ontology in 3-space at all is far, far weirder than "holism" or a mere violation of kinematical locality. These latter terms suggest (wrongly in this case) that we've got a basically sensible ontology of particles (or some elementary "subsystems") in ordinary physical space, but then there are properties of joint systems that aren't reducible to the properties of the individual systems composing them. Anybody who thinks ordinary QM is only that weird should have to explain in detail what the ontology is, exactly.

* This is a complete tangent now, but "marvellous point" is a slightly-barbed name for Albert's idea that the way to correctly understand dBB is in terms of everything playing out in 3N-dimensional configuration space: the ontology is a wave in this space and a single particle (the "marvelous point") being pushed around by the wave in this space. So the idea is that "really" or "fundamentally" reality is just a single particle being pushed around by a wave but in a very high-dimensional physical space. The appearance of many particles in 3-space is then somehow emergent from this. Now I think all other dBB fans think this is crazy (though Valentini seemed to endorse it once...??) -- it being considered crucial to a correct understanding of dBB that there are many particles in 3-space, whose configurations can account for our perceptions of trees, cats, pointers, planets, etc. My point in bringing it up was just that, despite disagreeing with Albert that this is a good way to understand dBB, I think he takes "kinematical locality" bull by the horns in a way that most don't. The wave function of a many particle system really just cannot sensibly be understood as describing stuff in 3-space, but with some surprising kind of "holism". The actual situation is much worse/weirder than that, I think.

I find Bell's terminology, distinguishing local from nonlocal beables, to be much more clarifying here than trying to think in terms of kinematical locality (and whether or not it is violated). I think these are in some sense two different attempts to get at the same basic issue, but the latter doesn't do it very effectively because, so to speak, it cuts too far from the joint. It presumes that the "worst case scenario" is far more benign than the actual situation we are confronted with in QM, and hence actually prevents people from grasping what is bizarre in QM.

Travis

Dear Robert,

I think you are right: There is an ambiguity about how to make the separation between kinematics and dynamics.

A somehow dramatic case is the relativistic kinematics. It is not only compatible with the wave-like version of the constancy of light as it is stated in the second postulate of SRT, but it is also compatible with a particle-like version of the constancy of light c.

Reference: Stachel, John. Einsteins Light-Quantum Hypothesis, or Why Did not Einstein Propose a Quantum Gas a Decade-and-a-Half Earlier? (Einstein. The formative Years, Einstein Studies 2000, p. 240)

In his 1905b-paper Einstein has noted that the velocity of light V cannot be altered by composition with any subluminal velocity.

This kinematical notion - if connected with the particle model of light - implies a far-reaching consequence : Even if the speed of light depends on the speed of the emitting source, the speed of light is always measured of being constant.

Consequently, all measurements concerning the second postulate of SRT are not unambiguous. There is - at least in principle - the possibility that something different has been measured - a sort of a particle-like version of the constancy of light.

Good Luck for your Essay.

Kind Regards

Helmut

Hi Rob!

I was happy to see that you've also entered this competition. The last time we saw each other I didn't get time to hear about your new insights. Now I get a chance to read about them!

Very thought provoking essay. I must say that I endorse the idea that causal structure should be fundamental. In fact, in shape dynamics we've reached the same conclusion through different reasoning. The argument is simple: we question the physical meaning of scale which forces us to do away with the conformal factor of the metric. The information that is left can be mapped one-one with the causal structure of spacetime (there is a theorem for this). However, the problem that we've been facing is that causal structure alone doesn't seem to be enough to recover the dynamics of gravity. I believe that something similar happens in the causal set approach.

So what could be wrong? I find your statement about the interchangeability of kinematics with dynamics persuasive but to difficult to make rigorous in general. However, in gauge theory one has complete control over this. Gauge theories are defined on a fibre bundle but the physics is only described on a particular section of this bundle. Each section represents a different kinematics but the dynamics can be accommodated in a precise way in order to describe the same physics. However, even in this rigorous setting there is a problem with the interchangeability of kinematics and dynamics: there can be global properties of the fibre bundle itself that can show up in the physics. An example of this is the Aharonov-Bohm effect that would be relevant to your model of the electromagnetic fields vs potentials. Similar things, like boundary terms, can appear during Legendre transforms which imply real differences between the Lagrangian and Hamiltonian. The chiral anomaly is yet another example relevant to the Standard Model.

Thus, I am sympathetic to your general argument but I wonder how these global effects, which are ultimately holistic, may also be important. One thing I have in mind, in particular, is the Weyl anomaly, which occurs when you try to gauge the conformal factor of the metric (i.e., when you try to remove all but the causal structure). I'm starting to believe that this may ultimately be related to origin of, let's call it "succession". In any case, I think there is some evidence that there might be some subtle structure in addition to causal structure that may be necessary to fully describe reality. Don't get me wrong, I would prefer it if this wasn't the case!

Take care,

Sean.

Robert

I applaud causal structure denouncing distinctions between kinetics and dynamics. But you must forgive me for already testing this unknowingly and proving it's worth, the results reported my essay.

I agree and indeed show the consequences of lambda (but also its derivative frequency) changing and the wave function being conserved with causality on all transformations. I'm not a quantum physicist so my language is different (even theatrical!) but the emergent causal structure of reality is self apparent, deriving classical relativity from a causal quantum mechanism. I hope you can read and comment for me. The physics is flooding out rather uncontrollably at present and needs help. My essay (to mix metaphors) is the tip of the iceberg.

Have you looked at the structure of truth functional logic with respect to compound systems. I only refer to dynamic logic (PDL) in my own essay but it is based on the hierarchical structure of 'nested' kinetic states with similarities to the infinite multiple components of ontic states, each only initially definable wrt the 'next state up' (local background state)?

As for your own essay I found it excellent. Clear, well argued, well written and correct, so a good score certainly coming. I hope you may feel the same of mine, if very differently presented.

Very best wishes for the contest.

Peter

[adapted from an email exchange with the author]

Hi Rob,

I just read your paper, and I found it beautiful.

I presented it in our group seminar, and some questions arose:

1) Your notion of kinematic locality on page 4: [math]\lambda_{AB} = (\lambda_A , \lambda_B).[/math]

You say that this is the same of separability, but it looks an awful lot like product states.

2) Newtonian physics.

Some people insisted that the kinematics, as they learned it, should be [math]\{q_i, \dot{q}_i\}_i[/math] (with the velocities), and not just the positions [math]\{q_i\}_i.[/math]

3) Practicability.

Is it always clear how to compute the causal-statistical parameters of a theory? For instance, how are they in your two examples (Hamiltonian and Newtonian physics) ? An expansion of the equations of motion?

4) Second page, your methodological principle vs operationalism

The definition of operationalism here wasn't super clear to me. After re-reading, I'd guess you take it to mean:

"make only claims about the outcomes of experiments, and not about the underlying reality."

Is this correct? The Plato Cave example illustrates what you mean by your principle very well, but I was left without understanding what kind of theories would fit operationalism in the example.

I have a guess:

The claim "shadows grow in the afternoon" (assume there is a concept of time and they call the hours before the dark "afternoon") respects operationalism.

The claim "shadows grow in the afternoon because there is a source of light sinking" does not respect operationalism, because it makes a claim about something (the source of light) that you cannot measure. It would however fit your methodological principle, because it helps explain something empirical.

Is this right? Is that why the 3D shape theory is not operational(ist)?

[also adapted from our email exchange]

Hi Lydia,

Nice to hear for you, and thanks for the comments.

1) This is a confusion of terminology, which I'm sorry I didn't clarify in the paper. The term "separability" is used by quantum information theorists to describe quantum states that are convex combinations of product states. In quantum foundations, the same term is sometimes used to describe an assumption about ontological models, namely that the ontic state space satisfies kinematical locality. You're right that lambda_AB is just the ontic state for AB. The consequences of kinematical locality on the epistemic states is just that we can write P(lambda_AB) = P(lambda_A,lambda_B) and we can therefore talk about whether lambda_A and lambda_B are correlated or uncorrelated, etcetera. Kinematical locality does not imply the quantum information theorists' notion of separability.

2) The information that needs to be specified to make predictions is certainly the positions and the velocities, but I don't think one should consider the velocities to be part of the kinematics. Maybe this argument will clarify why I think so: in a variational approach to classical mechanics, one could specify the initial position and the final position and deduce the trajectory followed by the particle in the intervening time. But one would not thereby conclude that the kinematics included the initial and final positions (at least, that's not how people usually talk about kinematics). So one shouldn't, I think, identify the variables used for boundary conditions with the kinematics.

3) The bit where I present the causal diagrams for Hamiltonian and Newtonian mechanics shows that one can easily translate a theory from the kinematical-dynamical paradigm into the causal paradigm. Deterministic dynamics is represented by a conditional probability distribution which is a point distribution on the conditioned variable for every value of the conditioning variable. For instance, in the Hamiltonian scheme, the conditional probability P(p2|q1,p1) is just delta(p2,f(q1,p1)) where delta(.,.) is the Kronecker delta and f(q1,p1) is just the function that defines p2 in terms of the earlier phase space point. That being said, these causal diagrams don't yet capture all and only the nonconventional bits. I'm not exactly sure what mathematical formalism does this. People in machine learning have introduced the notion of an equivalence class of causal diagrams, and this strikes me as promising.

4) As I see it, an operationalist is a kind of empiricist. Empiricism in the philosophy of science is the idea that the goal of science is simply to "reproduce the phenomena", for instance, to provide an account of what we experience. We should not ask "why", according to the empiricist, only "how". Empiricists were motivated to build knowledge on top of statements about experience because they thought that in this way it would be immune from error. This motivation was later convincingly shown to be misguided by people like Popper and Quine but in physics we still have a strong empiricist streak in our attitude towards quantum theory. The operational brand of empiricism is that the primitives in terms of which experience is described are experimental operations.

So, yes, "not about the underlying reality" is a good description of operationalism. If you look at any of the recent work on operational axioms for quantum theory, you'll get a feeling for the operational interpretation. Basically, an operationlist talks about preparations, transformations and measurements of systems, not about properties of systems or evolution of those properties. Your example of shadow growth is spot on.

The first couple of sections of this short paper that I wrote with Lucien Hardy describes in more detail the difference between realism and operationalism.

Hi Robert,

I really enjoyed your essay. It's a fascinating idea that kinematics and dynamics might be two aspects of a more fundamental causal structure. It leaves me wondering *why* causal structure should be so fundamental?

As I touch on in my essay, theoretical developments in understanding black hole physics, such as the holographic principle and horizon complementarity, seem to suggest that reality is radically frame-dependent, where frames are delineated in terms of causal structure. Do you suspect that this frame-dependence might shed light on the foundational role of causal structure?

Thanks, and again, I really enjoyed reading your work!

All best,

Amanda

Hi Rob,

Always good to hear from you too! Thanks for your detailed comments.

It's encouraging to hear that we seem to be coming to similar conclusions from the same philosophical motivations but different physical problems. I think it points to the strength of the principle that: "empirical indiscernables are physically identical" (not that "snappy" I have to admit!!). I've often thought of this as the core idea in Mach's principles but haven't been able to come up with a catchy slogan either. Regarding Leibniz, I don't think he explicitly mentions scale (or Mach either) but you should really check with Julian, who could tell you for sure.

The theorem I mentioned is one of the primary motivations for Causal Set theory, so Rafael could probably give you an exhaustive list of references (and he could probably explain the theorem more carefully than me). However, I think the original result was in: Journal of Mathematical Physics, July 1977, Volume 18, Issue 7, pp. 1399-1404. There are probably more modern versions though. I think there is a discussion and proof of this in Hawking and Ellis.

Regarding global structures like anomalies, boundary terms, and the AB effect. It would be interesting to see if one could reproduce these effects through a restriction of knowledge. Maybe this is naive, but wouldn't this restriction itself be just a kind of replacement for holism in the sense that it acts as a kind of global restriction on the system? In any case, I am coming to think more and more that these kinds of effects might be very fundamental.

Cheers,

Sean.

I just read your essay. I need to read it again for greater depth. I usually give these a couple of readings. I am pondering what you are saying. I think it comes down to the Bell statement

P(XY|ST) = sum_λP(X|Sλ)P(Y|Tλ)P(λ),

where the ontic variable is ultimately a summation or dummy variable. This means that what ever the analyst assigns to λ it is independent of the constituent subsystems. Your further implication is that any assignment of ontic variables for the constituent systems is then observer dependent and physically irrelevant.

In my essay I conclude something related to this. I remove the notion of locality from field theory. This is different from the statndard notion of locality, where in QFT this is commutivity of fields on a spatial amplitude fixed by a coordinate condition. However, for noncommutivative spacetime, say quantized spacetime or fine grained detail on a D-brane, quantum nonlocality occurs with gravity and this QFT construction is no longer afforded. The result, based on a reference of mine in this essay, is that the configuration variables of particles are gauge-like dependent. This means the universe consists of only one of each type of elementary particle, such as there is only one electron. The multiplicity of electrons, from the electron kicked up in energy by photosynthesis of one of my tomato plants, one electron in your computer passing through a logic gate, or one in a white dwarf exerting its degenerate pressure to hold the star up, and all the others, are due to a gauge dependent holographic projection of that one electron onto what we think of as local configuration variables.

In this setting the standard QFT approach to assigning field variables φ(x_i,t) at points on a spatial manifold is an observer dependent procedure which is not fundamentally correct. Consequently what you appear to advocate may have deeper implications with quantum gravity.

I am not well read on the subject of causal sets. However, this does suggest that the ontic variable is a sort of gauge-like "processor" that structures a network between nodes of events in the universe.

Cheers LC

Dear Robert,

In related to the hidden variable theories you mentioned in the essay, I have a different thinking that I hope you may find it interesting. Nothing mathematical fancy, I find that the bosonic quantum field can be reconciled from a system with vibrations in space and time. The model has some unique features that seem to be extendable to gravity and non-locality of quantum theory.

Is there really no reality in quantum theory

Your feedback will be valuable.

Hou Yau

Dear Sean and Robert,

When you make that important statement (Sean):

"However, even in this rigorous setting there is a problem with the interchangeability of kinematics and dynamics: there can be global properties of the fibre bundle itself that can show up in the physics. An example of this is the Aharonov-Bohm effect that would be relevant to your model of the electromagnetic fields vs potentials. Similar things, like boundary terms, can appear during Legendre transforms which imply real differences between the Lagrangian and Hamiltonian. The chiral anomaly is yet another example relevant to the Standard Model."

Does it follow that dynamics is to be preferred to kinematics where there is that departure? In other words, kinematics has a larger set of constraints and may not be as malleable in conforming to global properties and boundary terms. A case in point is the Aharonov-Bohm effect you mention which is not readily modeled by U(1) topologies. Relativistic kinematics seems to require a force fit into any even marginal compliance with SU(n) topologies.

Steve Sycamore

Hi Steve,

That's a good question. I don't have a good answer but I will think about it.

It may be that these global things - like global anomalies, boundary terms, and the non-trivial topology of the gauge bundle - should have their own status in a theory. In my view they play an important but mysterious role in physics. It may also be that Rob is right: that we can just accommodate them through a restriction on knowledge. My guess is that this restriction would have to be of a slightly different nature but I could easily be wrong!

Cheers,

Sean.

This is a very insightful essay!

It might be worth mentioning that computer scientists have long recognized the fact that kinematics and dynamics do not have separate observational meaning. This can be formalized in terms notions like bisimulation. There ought to be a definition of bisimulation for theories of physics!

Dear Robert! Excellent in-depth analysis in your essay. But why not have the depth of the ontology? Modern physics and mathematics - science not ontologically grounded. Today dominate the operational theory without the necessary ontological foundation.

Correctly, you have to look deep first structure. Bourbaki is a good title "mother" or "generative". Now, as for the physics and mathematics necessary first structure-mother. The concept of "causality" - a category of relationship (Immanuel Kant) ... What are the most fundamental relationship that is unconditional?

There is a fine principle of "coincidence of opposites." Also there is another good old principle of the triunity, who was not Isaac Newton.

In my absolute generating structure to form the basis of dialectical triad only absolute (unconditional) state of matter. This structure, which Umberto Eco calls "missing." "The truth must be drawn ..." (A.Zenkin) ... and ontologically grounded .... There is no other way to truth... My rating-9. Sincerely, Vladimir