On the Foundational Assumptions of Modern Physics by Benjamin F. Dribus

Ben,

The best thing about these contests is they give an opportunity to brain storm on a range of possibilities. The questions concerning physical foundations, particularly with respect to cosmology and quantum gravity, require different ways of thinking. I sometimes think that our educations have a disservice. While of course a graduate student needs to know classical and quantum mechanics, electromagnetic fields and so forth, I sometimes think these cement in our thinking so as to prevent successful consideration of deeper problems.

I sometimes think that we often suffer from some of the difficulties seen in elementary students with basic mechanics and F = ma. Our brains are predisposed to thinking in certain ways, and though we may learn the breakthrough physics of the past, this learning often serves to foster thinking that is erroneous on deeper foundations.

Time evaluated from the Jacobi variational principle

δt = sqrt{m_iδx_iδx_i/(E-V)}

is related to a proper time, or an interval. I might then say that if we multiply by E-V on both sides we get

(E-V)δt = sqrt{m_iδx_iδx_i(E-V)}

where the left hand side appears to be a Lagrangian times an interval of time. This may then be written as

∫d^3 δt sqrt{-g}R = sqrt{m_iδx_iδx_i(E-V)}

We may then break out the Ricci scalar R = R_{ab}g^{ab} and the left hand side exhibits this symmetry. On the right hand side again there is symmetry with the interchange of δx_iδx_j δ_{ij}. This probably needs to be firmed up of course, but I think this captures the idea.

Causal dynamics on the other hand is ordered by events with the idea of building up geometry. So there are orderings such as x < y so that in some product we have xy = -yx. This seems to have some connection with Penrose tensor space theory, where for every symmetric tensor there is an antisymmetric tensor. The relationship between the two is a graded algebra similar to supersymmetry. The symmetric interchange between spatial coordinates in shape dynamics is similar to the symmetric interchange between boson fields. The antisymmetric interchange of events in causal sets is similar to the interchange between fermions ψ(x)ψ(y) = ψ(y)ψ(x). Hence a causal set is potentially identical in form to a Slater determinant. This then opens the door to a type of functor or category theory which maps elements of geometry to elements of field theory.

Fields on a Cauchy surface separated by spatial intervals define the "shapes." Intervals separated by null or timelike intervals define causal sets. The first of these is symmetric, while the next is antisymmetric. This is similar to Penrose's tensor space, which axiomatizes spaces. If you have a space in n dimensions one can represent the positive tensor dimension as ||| ...|•ε = 0, where | represents an element such as a vector or spinor and the set |||...| means an exterior product of these. The ε means a Levi-Civita symbol and this is a skew product. This can be seen equivalently as a skew symmetrization of the |||...| in a higher dimensional space. If this is zero, then the space of tensors is symmetric. This system however requires there to be the |||...|•g, where g is a symmetric tensor. Again this is equivalent to a symmetric trace in a higher dimensional space. The "dimension of these tensors" are n and -n respectively. They correspond to the symmetric and antisymmetric sets of tensors, which have a duality.

This duality between symmetric and skew symmetric elements, or for two tensors products of the sort

{ψ^a, ψ^b} = g^{ab}

[φ^a, φ^b] = ω^{ab}

involves supersymmetry. In the case of spacetime the generators of supersymmetry Q_a and \bar-Q_b construct Lorentz boosts

{Q_a,\bar-Q_b} = iσ^μ_{ab}∂_μ.

where the momentum boost operator p_μ = -i∂_μ constructs the Lorentz group. Meanwhile {Q_a,-Q_b} = 0. The anticommutator of the super generators seems to have a categorical relationship with the antisymmetry of causal nets. The rotation operator M^{μν} and the super-generator obey

[Q_a, M^{μν}] = 1/2σ_{ab}^{μν}Q_b,

and the commutator between the momentum p^μ and the generator Q_a is zero

[Q_a, p^μ] = 0

The relationship between the symmetry and antisymmetric approaches, say shape dynamics and causal set theory, might then have functors to Fermi-Dirac fields and boson fields, and a system which includes both might then have a graded Lie algebra with Grassmann generators that connect the two.

Cheers LC

Lawrence,

Not to butt in here and while we have discussed this before, I usually avoid getting in the conceptual ring with you, but;

You make the two observations that prior knowledge can blind our thinking and that physics treats time as an interval.

That goes to my repeated observation that by treating time as a measure, physics only re-enforces the effect of sequence, rather than considering the cause of change, that is action. That it is not the present moving from one frame to the next, but action replacing configurations of the same material. Not the earth traveling a fourth dimension from yesterday to tomorrow, but tomorrow becoming yesterday because the earth rotates.

Duration doesn't transcend the present, but is the state of the present between observations, so there is no physical extension, only action.

If time were simply a dimension consisting of those intervals, wouldn't a faster clock rate move into the future more quickly, yet the opposite is true, as it ages/burns quicker, it moves into the past faster. Witness the twin in the faster frame has died, when her twin in the slower frame returns.

I feel like a frog on the road when I make this point, but while it only seems to be ignored, no one bothers to refute it.

Ben & John,

When it comes to time, I have no particular objective concerning its ontology. I find the idea that one want to can remove time and then say that objects in motion move through space with a velocity which we interpret as time as just another way of defining time. I find there is a sort of epistemological "dog chasing its tail" issue going on here. Space and time may exhibit a quantum uncertainty, where with quantum black holes if you measure a coordinate with arbitrary accuracy you lose all measurement of time and visa versa. I illustrate below how one can show ΔrΔt ~ 2Għ/c^4 = L^2_{Planck}/c with black hole physics. I tend to think that if one is going to "thump on time" then you have to thump on space as well. This is in part of why I think there is this curious duality between shape dynamics that plays with space and causal sets which play on time.

You might want to check out Hestenes, who is emeritus at U Arizona. He has been working a lot on figuring out how we can teach physics better. He cites examples about how students can take first year physics and learn to work the problems in the texts well enough to pass the course. Then when asked various conceptual questions about how something will move or behave under a force etc they completely get things wrong. He also makes a point that attending lectures and focusing in on them is itself a learned talent. We don't normally learn this way, and if you think about indigenous people young people learn by experience with their elders and by imitating them. I know that it was not until I was into college when I could actually focus on most of a lecture. Even still there were down times during lectures.

Of course now compound that with our digital age, which is duplicitous in my opinion. While there is much information available, much of our technology is meant to force us to think in tiny time frames. From 140 Twitter characters to split second video games I suspect the way our brains are being dendrite wired is different from the past, even the very recent past. Here you have a generation of young people moving up who have been acculturated by the digital age to think in split frame/time modes, rather than focusing in on something for long periods of time. We then wonder why young people have trouble in school, particularly boys. In fact my daughter has done very well in school, but my son is struggling horribly. He is not dumb either, but the whole classroom, lecture, book study and homework routine is just outside his personal ken.

The complementarity of space and time and the quantum mechanics of black holes could have had a much earlier start. At the 1930 Solvay conferences Neils Bohr and Albert Einstein debated the nature of quantum mechanics. Einstein was convinced of reality and locality and argued staunchly for an incompleteness of quantum mechanics. Quantum theory could only be made complete if there are some hidden variables that underlay the probabilistic, nonlocal quirky aspects of quantum mechanics. Einstein proposed an interesting thought experiment. Einstein considered a device which consisted of a box with a door in one of its walls controlled by a clock. The box contains radiation, similar to a high-Q cavity in laser optics. The door opens for some brief period of time t, which is known to the experimenter. The loss of one photon with energy E = ħω reduces the mass of the box-clock system by m = E/c^2, which is on a scale. Einstein argued that knowledge of t and the change in weight provides an arbitrarily accurate measurement of both energy and time which may violate the Heisenberg uncertainty principle ΔEΔt~ħ

Bohr realized that the weight of the device is made by the displacement of a scale in spacetime. The clock's new position in the gravity field of the Earth, or any other mass, will change the clock rate by gravitational time dilation as measured from some distant point the experimenter is located. The temporal metric term for a spherical gravity field is 1 - 2GM/rc^2, where a displacement by some δr means the change in the metric term is \simeq~(GM/c^2r^2)δr. Hence the clock's time interval T is measured to change by a factor

T-- >T sqrt{(1 - 2GM/c^2)δr/r^2} ~ T(1 - GMδr/r^2c^2),

so the clock appears to tick slower. This changes the time span the clock keeps the door on the box open to release a photon. Assume that the uncertainty in the momentum is given by the Δ p ~ ħ/Δr \lt TgΔm, where g = GM/r^2. Similarly the uncertainty in time is found as ΔT = (Tg/c^2)δr. From this ΔT > ħ/Δmc^2 is obtained and the Heisenberg uncertainty relation ΔTΔE > ħ. This demands a Fourier transformation between position and momentum, as well as time and energy.

This holds in some part to the quantum level with gravity, even if we do not fully understand quantum gravity. Consider the clock in Einstein's box as a black hole with mass m. The quantum periodicity of this black hole is given by some multiple of Planck masses. For a black hole of integer number n of Planck masses the time it takes a photon to travel across the event horizon is t ~ Gm/c^3 ~ nT_p, which are considered as the time intervals of the clock. The uncertainty in time the door to the box remains open is

ΔT ~ Tg/c(δr - GM/c^2),

as measured by a distant observer. Similarly the change in the energy is given by E_2/E_1 ~= sqrt{(1 - 2M/r_1)(1 - 2M/r_2)}, which gives an energy uncertainty of

ΔE ~ (ħ/T_1)g/c^2(δr - GM/c^2)^{-1}.

Consequently the Heisenberg uncertainty principle still holds Δ EΔT ~ ħ. Thus general relativity beyond the Newtonian limit preserves the Heisenberg uncertainty principle. It is interesting to note in the Newtonian limit this leads to a spread of frequencies Δω ~ sqrt{c^5/Għ}, which is the Planck frequency.

The uncertainty ΔE ~ ħ/Δt larger than the Planck mass gives an event horizon. The horizon has a radius R ~ 2GΔE/c^4, which is the uncertainty in the radial position ΔR associated with the energy fluctuation. Putting this together with the Planckian uncertainty in the Einstein box we then have

ΔrΔt ~ 2Għ/c^4 = L^2_{Planck}/c.

So this argument can be pushed to understand the nature of noncommutative coordinates in quantum gravity.

Cheers LC

Hello Ben,

congratulations, your essay is even more a charming tour the force through all concepts of the last 100 years then mine !

I like the Dribus Razor rejecting anything for a universal Schrödinger equation combined with Emmy Noether's conservation law in causal configuration space.

But be careful with rejecting the manifold structures the flaws may enter and you'll get lost without Plato's order of the heavens, means the dimensionality fo space.

Best wishes,

Renate

Ben,

Actually it started as a simple relativistic observation as to whether it is the present moving from past events to future ones, or the events going future to past. Basically I was applying a similar premise to the question of geocentric, vs. heliocentric. Whether it is the sun moving east to west, or the earth rotating west to east. I found, when considering it at length, that it gives a very different, inherently dynamic, view of reality, than the block time, static modeling that arose from assuming time is sequence and treating it as a measure of interval.

The difference between cause and effect and time is that sequence isn't cause and effect, but energy transfer is. Yesterday doesn't cause today, any more than one rung on a ladder causes the next. The sun shining on this spinning planet creates the effect called 'days.' Just as me tapping on these keys causes letters to appear on the screen, because there is some transfer of energy.

I did put the idea to Jonathan Kerr, but the first time, Aug 7, he was busy and the second time, Sep 12, he was irritated at my simplistic presumption and after some back and forth over the issue of clock rates, basically told me I was out to lunch. If you wish to consider the conversation on his thread, I'll let you be the judge.

My essay does go into psychology a bit, for the very good reason that sequence is the basis of personal experience. Us walking along that path, climbing that ladder, progressing through the days. So it is natural to build a theory of reality with it as foundational.

I used an earlier version as my entry in the nature of time contest, in 08.

Ben,

Hi. For my website, maybe try:

https://sites.google.com/site/ralphthewebsite/

and then click on the third link which is called " Why do Things Exist and Why is there something rather than nothing?" The section related to my point 5 was the last one before the conclusion; although, it helps to read the whole thing.

On the dimension thing, I would just say that for "events" or "structure is at the fundamental level" or geons or causal sets or anything to physically exist, which I think they have to if the universe is to be made of them, then I personally have trouble imagining how they could physically exist without having 3 dimensions. If one of the three is actually zero, then does this really exist in the non-mind, physical universe?

Anyways, no need to reply on this since it's good to have some differences in opinion (we don't want no groupthink!) and I know it kind of takes some time! Good luck at school and in the contest!

Roger

Dear Benjamin,

Thank you for reading my essay and giving good opinion. It is beneficial to me that one Ph. D. student gives good opinion about my essay. It is also a favor that someone reads it precisely and comments, this is what our amateur theories lack. So it is well to obtain any opinion, still better if it is professional or skilled one. It gives new ideas.

I hope that we will further exchange some physical opinions, also after this contests.

Because I cannot read all essays until Friday, can you recommend the best ones by your opinion.

I will read your essay tomorrow. I will give comment below of your essay.

Ben,

I thought I would repost this response to your observations :

I wouldn't describe time and sequence as purely mathematical, but as features of action. If I may use an example, it would be that time is frequency and temperature is amplitude. While one wave/cycle/step doesn't cause the next in the series, it does lead to it from the perspective of the dynamic manifesting the series. Cause is wholistic and the sum total cause of any event cannot be known prior to the event, because the lightcone(to use a spacetime concept) of input isn't complete until the event occurs. A bolt of lightening or bus might hit you before you make that next step and the energy manifesting you would be disrupted from its progression. It is just that we exist as a particular point of reference/one molecule of water in that tea kettle and so encounter a series of events within the larger dynamic.

As for entropy, it seems everyone always ignores that it only applies to a closed system. In an open or infinite system, energy lost by one system is gained by others. We are absorbing light that was radiated by other galaxies billions of lightyears away. It is only because the universe is presumed to be finite that it gains such prominence. Yet even in that model, this energy is simply being dispersed over an expanding area, not eliminated. I think the larger reality is a form of universal convection cycle of expanding energy and contracting mass. These galaxies drawing in mass and radiating energy, until that energy condenses back into mass and falls back into the closest galaxy.

You are quite right that "energy" defies clear definition, but think about that; Definition is structure and order and energy is constantly manifesting and dissolving structure and order. It's hard to put something in a box, when even the box is an aspect of what you want to put in it. Think of energy as what is physically real, whether radiant, potential, spin, attraction, repulsion, inertial. Even the absence of energy is a form of energy, in the vacuum. Energy manifests, information defines. Information arises from the interaction of different forms and degrees of energy. Such as that mass is a balance of positive and negative energies. We try to measure reality by banging energies into each other. Whether it is light from distant stars onto our telescopes, or ions in a particle collider. Or even cavemen banging one rock into another to see how it breaks/flakes.

Hi Ben,

I realize you wrote the above before the exchange we had that's posted with my essay, but I'd like to respond to your last comment.

I agree with your first points exactly -- the issue of observation is inherently very difficult, and also quite different from the types of issues physicists normally deal with. I certainly don't blame physicists for ignoring this issue as unlikely to be productive, even though QM makes it hard to ignore.

As to your last paragraph -- it's true that we have one "arrow of time", but it may not be as simple as we usually suppose. For example, the "acyclic" aspect of interaction that lets us conceive of time as linear does not yet distinguish a "direction" in time, but I think your causal structure does. So even with your primitive "binary relation" we have at least two levels of structure on a randomly connected interaction-web. And this doesn't yet give any definition of a metric, which I would guess may involve several kinds of structure. And then there's still more structure in the physical "present time" we experience, which involves the availability of "measurement-contexts" that can create new facts, and pass them on to participate in defining new measurement-contexts.

I would guess that the complexity of our interaction-rules is essentially mirrored in the complexity of spacetime structure, each rule contributing something different to making spacetime meaningful and measurable. It's just a holdover from the classical viewpoint that lets us imagine a simple, given spacetime background distinct from what happens in it. But I don't think of this in terms of multiple basic "binary relations" co-existing with each other.

I'd guess the causal-set structure defines very basic topological restrictions on the web of interaction, which provide a structural context in which certain higher-level constraints can define themselves, corresponding to other laws of interaction. Each level selects out part of the graph defined by previous levels, and gives new constraints that makes new selection-rules definable.

Incidentally, in case this conjecture makes any sense to you -- I have in mind your discussion of "covariance". I agree that symmetries in physics aren't fundamental -- I think of them as reflecting this kind of layering of information. Essentially a symmetry indicates a boundary between levels, where one kind of information is already definable but another is not. E.g. you get a circular symmetry if you can define a central point in a plane and the length of a radius, but angles and directions aren't yet definable. (I'm very vague about what the levels actually are, in physics, but given the way QM operates, I would guess that angles are more primitive than lengths.)

Needless to repeat, "there are several ways in which this reasoning could be wrong"!

Thanks -- Conrad

Dear Ben,

I read your essay. Your field being mathematics, as I expected, your remedy is also mathematical.In your comment about my essay, you mentioned about mathematical convenience. I agree. But convenience is more related to practical work than theory. Mathematical convenience may lead to (has already led to many) technology. The foundational question is more of theoretical nature, the question what matter actually is.

Let us suppose a lump of matter (whatever it may be) devoid of any motion. It will remain the same even after billions of years. So it is motion that causes changes. Motion is a space- time relation that can be mathematically stated, and any change can be mathematically interpreted. That is the role of mathematics.The cause- effect explanations based on mathematics need not necceesarily help us understand what that lump we call matter is. For that we require some physical assumptions about matter.

I have some assumptions about matter, and I think that it agrees with all observations, though it disagrres with the existing cause- effect explanations. However, there may be loop holes in that. So what I suggest is that the main stream scientists propose some physical assumptions about matter and verify whether these agrees with all the recorded observations.

Your suggestions regarding new mathematical approach and new experiments may sometimes lead to new technologies. The experiments can provide proof to the theory. But before that, you have to decide what physical assumption regarding matter is to be proved.