ChatGPT<p>Alright, let's break down the "All-at-once" Lagrangian models and retrocausality in quantum mechanics in a way that's easier to grasp. Imagine you're playing a video game that lets you control time—not just moving forward but also backward. This scenario will help us understand some pretty wild concepts in quantum physics.
Quantum Mechanics: The Basics<p>First, quantum mechanics is the part of physics that deals with really tiny things like atoms and particles smaller than atoms. It's famous for having rules that seem weird compared to the everyday world. One key idea in quantum mechanics is that particles can exist in multiple states at once until they're observed or measured. It's a bit like your video game character being able to take multiple paths simultaneously until you look to see where they actually went.
Lagrangian Models: Setting the Stage<p>In physics, Lagrangian models are a way to understand the dynamics of a system (like how particles move) by focusing on the path they take over time. Traditional physics often looks at forces and how they affect movement step by step. In contrast, the Lagrangian approach considers all possible paths a particle could take and finds the one that requires the least action—a concept involving energy and movement. It's like the game choosing the best overall path for your character, considering every single possible route from start to finish all at once, rather than deciding at each crossroad.
"All-at-once" Approach: The Plot Twist<p>The "All-at-once" Lagrangian models in quantum mechanics take this idea to another level. Instead of looking at events happening one after another (cause leading to effect, which leads to another effect, and so on), it looks at everything happening together, in a sense. The beginning and the end of a particle's journey are considered simultaneously, and the whole story is written in one go. It's as if the game not only chooses the best path but also considers how starting points and destinations can affect each other directly, ignoring the traditional flow of time.
Retrocausality: Time's Twist<p>Retrocausality is a mind-bending concept where an effect can precede its cause. Imagine if shooting an arrow at a target could make you decide to pick up the bow in the first place. In the context of quantum mechanics and those "All-at-once" models, this means the future can influence the past. It's like the game adjusting earlier parts of the journey based on what happens later, challenging our usual understanding of time.
Putting It All Together<p>The "All-at-once" Lagrangian models and retrocausality in quantum mechanics suggest that when we're dealing with the quantum world, the flow of time and the cause-and-effect relationships we're used to might not apply in the same way. It opens the door to thinking about physics and reality in a non-linear way, where the future and the past are part of a bigger, interconnected puzzle. This doesn't mean we can send messages to the past or change what has already happened, but it does challenge our perceptions and offers a fascinating way to think about the universe.<p>Just remember, these concepts are at the edge of our understanding and are still being explored and debated by physicists. So, if it feels a bit like science fiction, that's because we're still trying to figure out the science part!
I have a comment about Lagrangian models.<p>(I'm not commenting on the "All-at-once" angle, that is out of my league.)<p>You assert a contrast, with on one hand (traditional physics) tracking motion step by step, and on the other hand (Lagrangian) an approach that considers the overall path.<p>I will argue that in actual fact that contrast is far smaller than it appears to be.<p>In preparation I start with addressing the following: it is not the case that the true trajectory always coincides with a minimum of the action. There are also classes of cases such that the true trajectory coincides with a maximum of the action. Within the scope of Hamilton's stationary action there is an inversion: from classes of cases with minimum to classes of cases with maximum.<p>How can it be that within the single scope, here Hamilton's stationary action, both are viable?<p>The reason for that is: it is not about minimum nor maximum. The actual criterion is the property that the two have in common: as you sweep out variation: the point in variation space such that the derivative of the action is zero coincides with the true trajectory.<p>Next item in the preparation: the far reaching scope of differential equations.<p>When we solve a differential equation the solution that is obtained is a <i>function</i>. In that sense a differential equation is a higher level equation. A low level equation has a number as its solution. But a differential equation has an entire function as its solution. A differential equation states: this relation must be satisfied <i>concurrently</i> for all values of the domain. That is to say: when you solve a differential equation the solution that you obtain is for the <i>entire path</i>.<p>Now to the main point:
Calculus of variations has a particular mathematical property, I will use the catenary problem to showcase that property. The catenary problem: what is the shape of a chain that is suspended between two points? We consider the most general case: for any height difference between the two points of suspension. We have that the resting state is a state of minimal potential energy. That is to say: for the shape of the catenary the derivative of the potential energy wrt variation of the shape is zero.<p>Now divide the solution in subsections. Every subsection is an instance of the catenary problem. We can solve each of the subsections, and then concatenate those subsections. We can continue the subdividing; you can still concatenate the subsolutions. There is no lower limit to the size of the subsections; the reasoning remains valid down to infinitesimally short subsections.<p>Given that infinitesimal property: it follows that it should be possible to solve the catenary problem with a differential equation. I have on my website a demonstration of how to set up and solve the differential equation for the catenary problem. It's in an article titled: 'Calculus of Variations as applied in physics'.
<a href="http://cleonis.nl/physics/phys256/calculus_variations.php" rel="nofollow">http://cleonis.nl/physics/phys256/calculus_variations.php</a><p>More generally, this infinitesimal property explains why the Euler-Lagrange equation is a differential equation.<p>Action concepts are stated in the form of an integral, but here's the thing: the variational property obtains at the infinitesimal level, and from there it <i>propagates</i> to the level of the integral.<p>There is a note about the Euler-Lagrange equation (author: Preetum Nakkiran), in which the Euler-Lagrange equation is derived using differential reasoning only. That is: stating the integral is skipped altogether. That demonstrates that stating the integral is not <i>necessary</i> for deriving the Euler-Lagrange equation.
<a href="https://preetum.nakkiran.org/lagrange.html" rel="nofollow">https://preetum.nakkiran.org/lagrange.html</a><p>At the start of this comment I announced: the suggested contrast between traditional approach (force-acceleration) and Lagrangian approach is only an apparent contrast. On closer examination we see the two formalisms are in fact very closely connected.