Action principle: geometric and physical interpretation

[@gcarcassi]

These are results from our research project Assumptions of Physics. More details on this topic in our paper www.nature.com/articles/s41598-023-39145-y and out open access book: assumptionsofphysics.org/book/ . If you are interested in our active research, see our other channel www.youtube.com/@AssumptionsofPhysicsResearch

[@lukepierik2407]

Wow, this was a fantastic exploration weaving mathematical machinery with physical intuition. Thank you for your decades of persistence! 💯

[@gcarcassi]

Passing on the insights to people who appreciate it makes it all worthwhile!

[@Rey2u]

This video brings an amazing insight into the subject. The videos in this channel should be mandatory material in any advanced physics and engineering class.

[@justalittlestretch9404]

I just started watching your videos and it makes me want to take vacation days from work to go through them all. Something tells me this is so close to opening an even broader perspective. Maybe it is in your book or another video.

[@gcarcassi]

That would be our whole research project Assumptions of Physics... 😉

[@GeoffryGifari]

Hmm maybe there are things we can dig out deeper about this 1. Usually when the principle of stationary action is evoked, the initial point and the end point are specified, and the path is selected between two points which makes the variation of S = 0. I see that endpoints aren't necessary for this picture, we can have a line flowing forever and just define a surface 2. Can the value of action itself be defined in terms of the vector theta? 3. This one is in classical mechanics, but knowing that the action has a minimal (hbar) value in quantum mechanics, what kinds of insight can we make, using this method? (related to (2), the action itself sometimes pops up in quantum inside an exponential, so maybe there are new things we can find if it is written in theta vector) 4. Can the diagram incorporate special relativity? should energy be an axis ((ct,x,y,z) at equal footing to (E/c,px,py,pz)) while "Hamiltonian" is a component of the theta vector? would the constancy of the speed of light warp the diagram in any way? what about boosts? thanks!

[@gcarcassi]

1. Hhhhmm... If you get an infinite line with a variation, and get an "infinite strimp", I am not so sure you can apply Stokes' theorem in that case. If you can, then things should work the same. 2. Not sure I understand: the action is the line integral of the vector theta. So, yes, it is defined in terms of theta. 3. I am still figuring this out. In classical mechanics, the surface integral of S is connected to entropy. In quantum mechanics, the minimal uncertainty can be understood as a lower bound on entropy (I had another video on that). There should be a link, though I haven't found a way to tease it out. 4. Yes. This is another way to extend phase-space, so you get a full temporal degree of freedom. You need a minus sign in the energy/momentum four-vector. The Hamiltonian gets replaced by the "Hamiltonian constraint". This does two things: it is the generator for the evolution parameter and is a constraint on the overall system. For standard massive particles, the Hamiltonian constraint is the mass (expressed in energy) and the affine parameter is proper time. You also find "classical anti-particles" this way. You get also interesting parallel with Klein-Gordon and Dirac equations. That would be a whole other video series...

[@GeoffryGifari]

@@gcarcassi oh yes action in terms of vector theta is a bit clearer when i rewatched the video

[@UgoPiazzafan]

​@@gcarcassi Hello Gabriele, could you point a reference on the connection between entropy and the integral of action? And, in general, do you have written references for the results that you present in this video? :)

[@gcarcassi]

@@UgoPiazzafan What I talk about in this video is based on the pre-print linked in the comments. I do not have a reference for the connection between entropy and action - I have some ideas in my head, but I haven't done any literature research yet.

[@GeoffryGifari]

glad to have you back

[@gcarcassi]

You too! Thanks for your constant support!

[@maksim-surov]

4:50 It's not clear why surface Sigma must be tangent to the vector field S. Most likely Gabriele assumes a specific kind of variation, otherwise it's easy to define a "close" curve gamma' which twists around gamma, so the angle between the gamma' and S will be arbitrary. I will take a look at the article to clarify this.

[@gcarcassi]

Hi Maksim, with a finite variation, you can of course do whatever you want. The idea is that you are making an infinitesimal, first order variation of gamma. That is, take a point along gamma, take the direction along which you vary that point over gamma'. Now you have to vectors: one along gamma, and one that "connects" gamma to gamma'. Those two vectors identify a parallelogram, and that is the element dSigma along which you make the surface integral. Now that element is tangent to gamma because one side is actually the displacement along gamma. Does it make more sense?

[@GeoffryGifari]

hi gabriele, what do you think about the role of interpretation in relation to mathematical formalism in using a theory to explain natural phenomena? in which cases can the interpretation be "obvious" or not at all? how far can "bare math" gets us (or even just human language)? and how can the assumptions of physics project might play into this?

[@gcarcassi]

Not sure I can answer fully here... may need a full video. :-D So, mathematical "theories" do need "interpretation" in the sense that mathematical theories are just abstract symbols with relationships. Physical theories need a connection between mathematical symbols and measurements and predictions. This, at this point, I think can be done very well... more than I thought it could be done. There is a bit of reshuffling both on the math and the physics side (i.e. we may not have 100% the right math or the right physical concepts), but you can do it. I think a one-to-one dictionary between physical concepts and mathematical representation is possible and desirable. In the project Assumptions of Physics, we try and find exactly what elements are necessary and sufficient to get to a particular theory. However, in the context of quantum mechanics, interpretation try to do more than that... and that I think is problematic. Mainly, I don't think the questions that interpretations try to answer are ultimately illuminating on the physics side. (If you are doing philosophy is different) Note that by interpretations I really mean things that are completely equivalent to QM... Things like GRW I do not put in the same category (they have different predictions, they are different theories).

[@GeoffryGifari]

@@gcarcassi hmmm what do you think about the cases where mathematical expressions are derived from an existing theory first, then the work is to interpret it (ex: photoelectric effect -> light itself is quantized, negative energy solutions -> antiparticles, mathematical event horizon -> black holes, etc), compared to physical phenomenon is discovered first, then try to find the math (ex: multiple quark flavors -> SU(N) gauge theory, most of earlier classical physics findings)? Seems like one-to-one dictionary between physics concepts and mathematical representation can be very tricky, if we have to choose either from the space of possibilities btw if the replies are better laid out in a video (or even a series), it will be great

[@gcarcassi]

@@GeoffryGifari When you are in the process of discovery, you use whatever strategy... even ideas that are partially wrong. No issue there. But once a theory is consolidated, then it's the time to get the details in order.

[@RafaelSCalsaverini]

If a displacement field is not divergenceless, is it a dissipative system? If yes, it would be interesting to find out if you can create a formalism for (at least some classes of) dissipative systems using this framework and relaxing the zero divergence hypothesis.

[@gcarcassi]

Yes, non-divergence should mean either dissipative or driven (energy flowing in). The worst thing that fail in that case is that the flow does not depend on the contour anymore... so I do not see how to make the new "Lagrangian" be dependent only on the path.

[@tosifahamed]

This was wonderful. How would one add friction and time dependent forcing into this framework. I'm guessing it will break the reversibility assumption that you had. Is it still possible to generalize it in that case?

[@gcarcassi]

Thanks! It's not possible to generalize when you have friction. If the evolution is not deterministic-and-reversible, i.e. it's not divergence free, two things happen: the flow does not admit a potential, so you can't define an action, and the flow through a surface does not depend only on the boundary, so you wouldn't be able to define a single flow of states in the region between the path and its variation. Time dependence is different: there are some forces that, though they are time dependent, they are deterministic and reversible (i.e. conservative). You can generalize to that case by separating the use of time as an evolution parameter and time as a state variable (e.g. you have x(s) and t(s) where s is an affine parameter and t time as defined in a particular frame). What you get, without further assumptions, is relativistic mechanics! All these cases will be explored in the new version of our open access book (you can always find the latest draft at github.com/assumptionsofphysics/book )

[@GeoffryGifari]

oh, i was thinking: if Hamiltonian mechanics can be derived assuming determinism and reversibility, knowing that quantum mechanics incorporate *Hamiltonian operator* as the generator of time evolution, can we say that unitary evolution of the wavefunction is also intrinsically related to determinism and reversibility (while the probabilistic nature of QM comes after measurement, a non-unitary process and overall thorny issue)

[@gcarcassi]

Correct: unitary evolution is deterministic and reversible evolution over quantum states.

[@nzuckman]

Mathematical! 🤩

[@giveaquestion9683]

Hi Gabriele, I have a question regarding the claims that the gauge fixing of theta and the lagrangian is arbitrary (in the discussion), and that the derivative of the time parameter of the path with respect to time is always 1 ( eq. 3). The use of forms for the general case immediately made me think of the reletevistic generalization (even to GR, but let's start with SR). Wouldn't the requirement of Lorentz invariance lead to a choice of gauge that is 1) more natural and more restrictive in a way that will give better credence to the lagrangian and the action as physical significant quantities? and 2) will change the derivative of the time parameter in respect to time of the observer to gamma (from SR)? Best regards and thank you for the lecture, it was very interesting.

[@gcarcassi]

The answer may require another video... Let's see what I can say in the space of a comment. The gauge you see here fixes, among other things, the relationship between conjugate momentum, which is not uniquely defined, and kinetic momentum. The "most natural" gauge is where kinetic momentum is equal to conjugate momentum plus the vector potential of the magnetic field. If you want to make things relativistic, note that technically there are "two times". One is the coordinate t and one is the affine parameter for the curve \gamma(t). If you make a generic coordinate transformation, you mix the coordinates, but the affine parameters remains the same. So, to give a real relativistic treatment, you should use a different variable for the affine parameter. Which is an extra choice for the "gauge."The best expression, then, is to use proper time for particle and minus proper time for antiparticle. That's the "more natural" expression. All of this is easier to see if you extend phase space by both time and energy. If you do things properly, Lorentz invariance (i.e. a local Minkowsky metric for space/time) comes out by itself. I am trying to find a way to make the argument that requires the least math possible, but haven't finished yet. It still an open problem to me whether the curvature of the metric tensor is linked or not to the EM vector potential. Not sure if any of this is intellegible... it would be easier with equations and pictures. 😁

[@ameyodas9042]

really informative and in depth video! however, could you please clarify why the action (in case of least action, for say), can't be zero in general (a surface perfectly aligned to the displacement field/flow), instead of a minimum lower bound. how would that translate to the geometric interpretation? thanks!

[@gcarcassi]

I don't think I understand the question... The action can be zero, positive, negative. The variation of the action is what corresponds to the flow of states. This can also be positive if the surface between the paths "catches" flow in between.