How do you view Physics as a whole?

[jordi]

Usually it is stated that physics is divided among classical mechanics, classical field theory, quantum mechanics, quantum field theory and statistical mechanics, with hbar, the speed of light and the number of particles being the parameters differentiating all these theories.

However, despite intuitive claims, it seems that it is not obvious that CM is a limit of QM, or that QM is a limit of QFT. And even if they were, their empirical analysis (CM looks at trajectories, QFT looks at S-matrix) is so different, that it makes sense to differentiate them.

So, it seems then to "define an axiom of physics", stating that physics "is" the Standard Model and GR, and the rest are approximations of these two theories, in some limits, is problematic.

But at the same time, there is a clear "unity in formalism" (say, all these theories involve functions from the spacetime manifold to the configuration manifold, and empirical values can be obtained from correlation functions, which are calculated as some kind of path integral).

Is there a way of thinking about this issue that gives you "peace of mind"?

 

[TeethWhitener]

However, despite intuitive claims, it seems that it is not obvious that CM is a limit of QM, or that QM is a limit of QFT.

https://en.wikipedia.org/wiki/Klein–Gordon_equation#Non-relativistic_limit
That might get you started.

 

[jordi]

https://en.wikipedia.org/wiki/Klein–Gordon_equation#Non-relativistic_limit
That might get you started.

There are many arguments for this limit (Ehrenfest theorem, WKB approximation, hbar to 0 in the path integral ...) but I think the lack of spread of the wave packet is not easy to explain, in this limit. I have read (I do not know how certain this claim is) that it is not obvious that CM can be recovered from QM and / or QM can be recovered from QFT. They could be different theories.

 

[PeterDonis]

the lack of spread of the wave packet is not easy to explain, in this limit

There are no wave packets in the limit you describe, so there's nothing to spread.

I have read

Where? Please give a specific reference.

 

[jordi]

There are no wave packets in the limit you describe, so there's nothing to spread.
Where? Please give a specific reference.

https://aapt.scitation.org/doi/10.1119/1.4751274#:~:text=88.,x,t)=0.

 

[TeethWhitener]

The most important limit for going from QM to CM is probably $$\lim_{\hbar\to 0} [\hat{x},\hat{p}]=0$$
Here’s @samalkhaiat ’s PF Insight; it’s a really good resource:
https://www.physicsforums.com/insights/the-classical-limit-of-quantum-mechanical-commutator/

 

[PeterDonis]

https://aapt.scitation.org/doi/10.1119/1.4751274#:~:text=88.,x,t)=0.

It looks like this is the arxiv preprint, for those who can't get behind the paywall:

https://arxiv.org/abs/1201.0150

 

[vanhees71]

The author discusses the WKB approximation. It's an approximation you can make in general for theories related to the wave equation, and it is in fact the way how Schrödinger arrived at his equation: As the approximation of "wave optics" in terms of the eikonal approximation is "geometric optics", the eikonal approximation of the Schrödinger equation leads in leading order to the Hamilton-Jacobi partial differential equation which is equivalent to particle dynamics in mechanics.

Another point the author makes is also valid since indeed the equations for the expectation values of observables fulfilling Ehrenfest's theorem are not identical with the classical equations of motion. That's the case obviously only if the equations of motion are linear, i.e., in the few examples of (a) the free particle, (b) motion in a constant force field, (c) the harmonic oscillator.

What's not discussed is the case of macroscopic objects, which of course under the usual circumstances obey the laws of classical physics. This comes about because you describe the macroscopic relevant observables in a "coarse grained" picture, i.e., averaging over the quantum (and thermal) fluctuations, which leads to a description in some hierarchy of "coarse-graining steps" to the classical laws: Boltzmann(-Uehling-Uhlenbeck) transport equations as the Markov limit of the Kadanoff-Baym equations for one-particle Wigner function, enabling an interpretation of the corresponding coarse grained quantity as a phase-space distribution function for a fluid -> viscous/ideal hydrodynamics for systems close to or in local thermal equilibrium, etc.

 

[PeterDonis]

it is not obvious that CM can be recovered from QM

The paper you reference only talks about the limit $\hbar \to 0$. It is correct that taking that limit is not always sufficient to recover CM from QM. But that is not the same as saying that CM can't be recovered from QM at all; there might be other ways of doing it. For example, the Ehrenfest Theorem does not involve any $\hbar \to 0$ limit.

 

[vanhees71]

The Ehrenfest theorem, however says that
$$m \mathrm{d}_t^2 \langle x \rangle=- \left \langle \vec{\nabla} V(\vec{x}) \right \rangle.$$
This is usually not the same as
$$m \mathrm{d}_t^2 =-\vec{\nabla}_{\langle \vec{x} \rangle} V(\langle \vec{x} \rangle).$$
So the author of the paper is right in saying that this is usually not an accurate proof for the validity the classical limit.

It's of course one under circumstances, where the quantum fluctuations around the average quantities are small in some sense to be specified in the context of the considered situation.

 

[jordi]

The Ehrenfest theorem, however says that
$$m \mathrm{d}_t^2 \langle x \rangle=- \left \langle \vec{\nabla} V(\vec{x}) \right \rangle.$$
This is usually not the same as
$$m \mathrm{d}_t^2 =-\vec{\nabla}_{\langle \vec{x} \rangle} V(\langle \vec{x} \rangle).$$
So the author of the paper is right in saying that this is usually not an accurate proof for the validity the classical limit.

It's of course one under circumstances, where the quantum fluctuations around the average quantities are small in some sense to be specified in the context of the considered situation.

Sure, but how do you prove that these small quantum fluctuations never become macroscopically large? (as it happens in reality).

Also, there is another issue (maybe this one is well understood, and only my knowledge is faulty): CM observes the positions of objects. By observing how the Moon moves, we can be certain that CM is "right".

Instead, in QM, and especially in QFT, we usually observe S-matrices, so collisions of particles which are initially (and at the end) very far away of each other. True, quantum physics also analyzes successfully bound states. But I have the feeling (as said, maybe I am wrong) that the empirical consequences are very different in CM than in quantum theories.

 

[vanhees71]

Yes, and the moon behaves according to CM because of being a large (macroscopic body) and decoherence. Already the "permanent interaction" of the moon with the cosmic microwave background equation is enough to make its motion classical. It has not so much to do with the WKB approximation but rather with the fact that it is very difficult to keep a macroscopic object isolated from the environment. For a nice intro to this see

https://arxiv.org/abs/quant-ph/9506020

 

[samalkhaiat]

It looks like this is the arxiv preprint, for those who can't get behind the paywall:

https://arxiv.org/abs/1201.0150

It does not worth reading.

Dirac’s famous book [1] on quantum theory states that “...classical mechanics may be regarded as the limiting case of quantum mechanics when $\hbar$ tends to zero.” In quantum mechanics a single particle in an external potential is described by Schrodinger’s equation,
$$\begin{equation}\left[ \frac{\hbar}{i}\frac{\partial}{\partial t} - \frac{\hbar^{2}}{2m} \sum_{k = 1}^{3} \left( \frac{\partial}{\partial x_{k}} \right)^{2} + V (x,t) \right] \psi (x,t) = 0. \end{equation}$$
Thus, Dirac’s statement would imply that Newton’s second law,
$$\begin{equation}\frac{d}{dt}mr_{k}(t) = p_{k}(t), \ \frac{d}{dt}p_{k}(t) = − \frac{\partial V(x,t)}{\partial x_{k}}|_{x = r(t)} ,\end{equation}$$
should follow from Schrödinger’s equation in the limit $\hbar \to 0$.
Nobody has ever performed a general exact calculation showing that Eq.(1) implies Eq.(2) in the limit $\hbar \to 0$.

Read (or re-read carefully) Section 31 of Dirac’s “famous book” to realize that the underlined statement is false.

 

[Demystifier]

Is there a way of thinking about this issue that gives you "peace of mind"?

In my case, it's "Bohmian mechanics for instrumentalists" (link in my signature below) that gives me a peace of mind.

 

[A. Neumaier]

there is a clear "unity in formalism" (say, all these theories involve functions from the spacetime manifold to the configuration manifold, and empirical values can be obtained from correlation functions, which are calculated as some kind of path integral).

Is there a way of thinking about this issue that gives you "peace of mind"?

My way of viewing the unity of physics and the various limits is spelled out here. It gives a systematic view of physics classified in terms of 7 orthogonal criteria:

The first criterion is methodological, and distinguishes between

• applied physics (AP), didactical physics (DP), experimental physics (EP), theoretical physics (TP), and mathematical physics (MP).

The other six criteria are defined in terms of the six limits that play an important role in physics:

• the classical limit (ℏ→0) distinguishes between classical physics (Cl), in which ℏ
• is negligible, and quantum physics (Qu) where it is not.
• the nonrelativistic limit (c→∞) distinguishes between nonrelativistic physics (Nr), in which $c^{−1}$ is negligible, and relativistic physics (Re) where it is not.
• the thermodynamic limit (N→∞) distinguishes between macroscopic physics (Ma), in which microscopic details are negligible, and microscopic physics (Mi) where they are not.
• the eternal limit (t→∞) distinguishes between stationary physics (St), in which time is negligible, and nonequilibrium physics (Ne) where it is not.
• the cold limit (T→0) distinguishes between conservative physics (Co), in which entropy is negligible, and thermal physics (Th) where it is not.
• the flat limit (G→0) distinguishes between physics in flat space-time (Fl), in which curvature is negligible, and general relativistic physics (Gr) where it is not.

A particular subfield is characterized by a signature consisting of choices of labels (or double arrows between labels) in some categories.

A few examples:

• Thermodynamics: Ma ,Th
• Equilibrium thermodynamics: Ma, Th, St
• Classical Mechanics: Cl, Co
• Classical field theory: Cl, Co, Ma
• General relativity: Cl, Re, Ma, Gr
• Quantum mechanics: Qu, Nr
• Relativistic quantum field theory: TP, Qu, Re, Mi
• Statistical mechanics: TP, Mi<−>
• Ma, Th
• Precision tests of the standard model: TP<−>
• EP, Qu, Re, Mi, St, Co
• The empty signature is simply the field of physics itself.

 

[vanhees71]

There may be some difficulties of thinking about the approximation from quantum to classical mechanics in terms of $\hbar \rightarrow 0$ (i.e., singular perturbation theory aka WKB), but what's really making macroscopic objects appear classical is decoherence and the difficulty of isolating the classical object from "the environment".

Then I'm a bit puzzled about the statement that SR in the classical has foundational issues that are resolved by the generalization to GR. Can you explain this in a bit more detail or give a reference? I have no clue what you are talking about.

Of course, there are foundational issues with QFT since it's mathematically not strictly defined and used "only" in the sense of effective low-energy models, but whether there is a more comprehensive theory underlying it, is hitherto not known.

 

[RUTA]

I started looking into foundations of QM in 1994 and only this year found a way to pull physics together self-consistently. I have an Insight that I've been working on showing a particular deep unity between SR, QM, and GR via their "mysteries" and the fundamental constants, c, h, and G, respectively. The take-home message is simply that "no preferred reference frame" (NPRF) and the fundamental constants lead to their "mysteries" (time dilation and length contraction, Bell state entanglement and the Tsirelson bound, and the contextuality of mass, respectively). Essentially, we move to what Einstein called "principle explanation" a la the postulates of SR. To complete the picture we move beyond physics to answer the question, "What is physics about?" Here is that paper: “Re-Thinking the World with Neutral Monism: Removing the Boundaries Between Mind, Matter, and Spacetime,” Michael Silberstein and W.M. Stuckey. Entropy 22, 551 (2020) https://www.mdpi.com/1099-4300/22/5/551/pdf. If you're a physicist, you probably won't want to read all the consciousness stuff, so you can just skip to Section 3. This paper (still under review although the results have already been published by us and others) provides the details of Section 4.2: “Answering Mermin’s Challenge with Conservation per No Preferred Reference Frame,” W.M. Stuckey, Michael Silberstein, Timothy McDevitt, and T.D. Le. (2020) http://arxiv.org/abs/1809.08231. Here is a paper that explains the Tsirelson bound via NPRF: “Why the Tsirelson Bound? Bub’s Question and Fuchs’ Desideratum,” W.M. Stuckey, Michael Silberstein, Timothy McDevitt, and Ian Kohler. Entropy 21, 692 (2019) https://arxiv.org/abs/1807.09115.

In short, I believed there was a deep inconsistency between QM and SR when I started my study of foundations in 1994. That view is not uncommon, e.g., see quotes from Dirac, Bell, and others in "On the Incompatibility of Special Relativity and Quantum Mechanics," M. Mamone-Capria, Journal for Foundations and Applications of Physics 8(2), 163 (2018) https://arxiv.org/pdf/1704.02587.pdf. I have since come to believe they are deeply coherent, as spelled out in the above cited papers. I'll Submit the Insight "A Principle Explanation of the "Mysteries" of Modern Physics" today, as it provides an overview for the layperson.

 

[Sunil]

There are many arguments for this limit (Ehrenfest theorem, WKB approximation, hbar to 0 in the path integral ...) but I think the lack of spread of the wave packet is not easy to explain, in this limit. I have read (I do not know how certain this claim is) that it is not obvious that CM can be recovered from QM and / or QM can be recovered from QFT. They could be different theories.

In realistic interpretations like de Broglie-Bohm, Nelson and so on there is no such problem because there are also trajectories of the configuration itself. And for these trajectories the limit is easy.