This is the first post in my series about Quantum Field Theory. What a let-down: I will just discuss classical mechanics.

There is a quantum mechanics, and in contrast there is good old classical, Newtonian mechanics. The latter is a limiting case of the former. So there is some correspondence between the two, and there are rules that let you formulate the quantum laws from the classical laws.

**But what are those classical laws?**

Chances are high that classical mechanics reminds you of pulleys and levers, calculating torques of screws and Newton’s law F = ma: Force is equal to mass times acceleration.

I argue that classical dynamics is most underrated in terms of geek-factor and philosophical appeal.

**[Space Balls]**

This might have been ingrained in your brain: A force is tugging at a physical object, such as earth’s gravity is attracting a little ball travelling in space. Now the ball moves – it falls. Actually the moon also falls in a sense when it is orbiting the earth.

When bodies move their positions change. The strength of the gravitational force depends on the distance from the mass causing it, thus the force felt by the moving ball changes. This is why the three-body problem is hard: You need a computer for calculating the forces three or more planets exert on each other at every point of time.

So this is the traditional mental picture associated associated with classical mechanics. It follows these incremental calculations:

*Force acts – things move – configuration changes – force depends on configuration – force changes.*

In order to get this going you need to know the configuration at the beginning – the positions and the velocities of all planets involved.

So in summary we need:

- the dependence of the force on the position of the masses.
- the initial conditions – positions and velocities.
- Newton’s law.

But there is an **alternative description of classical dynamics, offering an alternative philosophy of mechanics so to speak.** The description is mathematically equivalent, yet it feels unfamiliar.

In this case we trade the knowledge of positions and velocities for fixing the positions at a start time and an end time. Consider it a sort of game: You know where the planets are at time t_{1} and at time t_{2}. Now figure out how they have moved / will move between t_{1} and t_{2}. Instead of the force we consider another, probably more mysterious property:

It is called the *action*. The action has a dimension of [energy time], and – as the force – it has all information about the system.

The action is calculated by integrating…. I am reluctant to describe how the action is calculated. Action (or its field-y counterparts) will be considered the basic description of a system – something that is given, in the way had been forces had been considered given in the traditional picture. The important thing is: You attach a number to each imaginable trajectory, to each possible history.

The trajectory a particle traverses in time slot t_{1}-t_{2 }are determined by the **Principle of Least Action **(which ‘replaces’ Newton’s law): **The action of the system is minimal for the actual trajectories**. Any deviation – such as a planet travelling in strange loops – would increase the action. It is like a particle tests all possible paths and calculates their associated actions. Near the optimum path the action does hardly vary.

**This sounds probably awkward – why would you describe nature like this?
** (Of course one answer is: this description will turn out useful in the long run – considering fields in 4D space-time. But this answer is not very helpful right now).

That type of logic is useful in other fields of physics: A related principle lets you calculate the trajectory of a beam of light: Given the start point and the end point a beam, **light will pick the path that is traversed in minimum time** (This rule is called Fermat’s principle).

This is obvious for a straight laser beam in empty space. But Fermat’s principle allows for picking the correct path in less intuitive scenarios, such as: What happens at the interface between different materials, say air and glass? Light is faster in air than in glass, thus is makes sense to add a kink to the path and utilize air as much as possible.

**[Baywatch]**

Richard Feynman used the following example: Consider you walk on the beach and hear a swimmer crying for help. Since this is a 1960s text book the swimmer is a *beautiful girl*. In order to reach her you have to: 1) Run some meters on the sandy beach and 2) swim some meters in the sea. You do an intuitive calculation about the ideal point of where to enter the water: You can run faster than you can swim. B*y using a little more intelligence we would realize that it would be advantageous to travel a little greater distance on land in order to decrease the distance in the water, because we go so much slower in the water* (Source: Feynman’s Lecture Vol. 1 – available online since a few days!)

Those laws are called variational principles: You consider all possible paths, and the path taken is indicated by an extremum, in these cases: a minimum.

Near a minimum stuff does not vary much – the first order derivative is zero at a minimum. Thus on varying paths a bit you actually feel when are close to the minimum – in the way you, as a car driver, would feel the bottom of a valley (*It can only go up from here*).

**Doesn’t this description add a touch of spooky multiverses to classical mechanics already?** It seems as if nature has a plan or as if we view anything that has ever or will ever happen from a vantage point outside of space-time.

Things get interesting when masses or charges become smeared out in space – when there is some small ‘infinitesimal’ mass at every point in space. Or generally: When something happens at every point in space. Instead of a point particle that can move in three different directions – *three degrees of freedom* in physics lingo – we need to deal with an infinite number of degrees of freedom.

Then we are entering the world of **fields** that I will cover in the next post.

Related posts: Are We All Newtonians? | Sniffing the Path (On the Fascination of Classical Mechanics)

Another professional on a quest to better grasp QFT posts at physics-quest.org His way of doing this is by writing an open QFT textbook. I only looked at his first chapter so far, but I think it looks rather promising, jumping right into the thick of things with Green function and propagator treatment of classical EM.

Thanks, Henning – my list gets longer and longer :-) I had a quick look now – yes, it seems a great idea to introduce the propagator early on – very different from introducing Noether’s theorem, Lagragians, canonical quantization, time ordering etc. first.

I also found this text book rather interesting: http://www.quantumfieldtheory.info/ The first chapters are available online – the author had also made them available online while he was writing the other chapters of the book.

I find these different ‘philosophical’ routes to QFT very interesting. I also have Zee’s book QFT in a Nutshell which starts from path integrals.

Actually, when I started this blog I had considered an ‘open physics notebook’ approach myself – similar to what http://physicspages.com/ does, but I finally decided to go for a ‘pop-sci’ approach in writing. Not sure how far I will get.

The strange thing is, I always feel I understand your physics posts. Then I read what science-minded readers write and I wonder if I’m delusional. I wouldn’t trust me to use any of this knowledge I glean from you, but everything I learn makes life more interesting, certainly richer.

Have you ever read the Dune series by Frank Herbert?

You are a geek at heart, Michelle! Don’t underestimate your physics grasping super powers! :-)

I am really interested by the way different people understand science. This whole series is an experiment. So thanks a lot for commenting!!

And, yes, I have read Dune. But it was a bit too epic, too much fantasy (Lord-of-the-Rings-style) for my taste.

I did refer to it as a fantasy when I began the comment, but Wikipedia classifies it as sci-fi so I rewrote it in my comment. There is a strange collapse of physics, philosophy, and psychology in that series. Your post left me thinking of them, especially the discussion of ‘the golden path’ and space travel. I don’t remember it well enough, but it seemed to connect with the idea of bending light to travel through space more quickly, and the consideration of multiple possibilities.

thanks for calling me a geek, btw, I will consider it a badge of honour. :)

I should probably read it again. I read it during holidays – I had picked it at the airport, about 20 years ago, and I read the German version. I need to check but if I recall correctly it felt victim to my most recent cleaning and disposal session :-)

Ah, then there’s no point going back to it. I was rereading the first book a couple of weeks ago and lost patience with it. I set it down without finishing, so I won’t insist in any way at all that you try to reread the books. But I did enjoy them the first time I read them, and they gave me lots to think about for quite some time.

Allright, you’ve made me sign up to WP ;-) But… I’ll strike back, promised. Be prepared. Until then: Carry on, it’s absolutely amazing.

Welcome to the WordPress-o-sphere Mr. Strummer ;-) :-) Looking forward to your counterstrike!

Well… Vacuum is a tricky business, yet – this might give you clue :-)

You are in the harvesting-the-Zero-Point-Energy-from-vacuum-perpetuum-mobile business now? Where can I sign up in order to invest?

a) not really – it’s more like running a flea circus… training electrons to behave as expected when passing through a vacuum tube.

b) unfortunately all shares have been sold, but we’re planning a second financing round, I’ll let you know in time

Oh my gosh Elke … you’ve lost me already. And, you promised to explain in such a way that even a biologist could understand. I read this post almost immediately after it was published and got confused … so I assumed I was tired or not in the correct mental state to take it in … so I tried again this morning. Argh. I have always wondered how it is that your mind is wired such that you ‘see’ this stuff so easily … and how mine is wired such that I don’t take in any more than the very surface. Perhaps I didn’t pay close enough attention when I took physics in college? But no … that sort of physics … highly dependent upon algebra (not calculus) … was easy, fun, and something I understood. What you are asking me to do here is beyond me! I want to understand but simply do not have the background. You really do need to do some posts which can be titled Quantum Mechanics for Dummies. D

Hi Dave! Thanks a lot for your feedback – I really do appreciate it! I want to know if I can get my message across to my target audience.

So, first of all – I am sorry that I lost you. I had the best intentions though: I am even started reading a book by a science writer who taught herself calculus (and who had no exposure to calculus before) to get ‘into’ science popularization mode. I revisited my high school physics books – trying to tune it to approximately that level.

However, I believe, years of slogging through advanced physics problems at the university have probably really re-wired my brain. Maybe I tend to translate anything back to math (incl. popular non-mathy science books). So the good thing is, these exercises have not been in vain and I didn’t lose that ability despite my long stint in the IT industry. The bad thing seems to be that I lost (or never had) the gift to explain it no an audience without a math-dominated science background.

But I had planned to write a more ‘philosophical’ post about math as a language in physics, referring to that said book. I will try again to explain the gist of how a physicist ‘sees’ classical dynamics (as this post had not yet been about anything quantum!).

As a biologist – do you / did you work with differential equations covering the feedback loops in systems in nature? Such as food chains, the proliferation of bacteria or chaotic behavior of changing weather?

All my talking about forces tugging at balls in space can be boiled down (mathematically) to similar equations. Last year I wrote about the simplest differential equation in physics and related concepts:

https://elkement.wordpress.com/2012/11/20/are-we-all-newtonians/

Probably I also tried to avoid to much repetition here although I wanted to make the current post a self-consistent one.

Tell you what … I will follow the link to the explanation of the differential equation and let you know if I understand … and we can go from there! Thanks for understanding my short-coming! I’ll get back to you. D

Thanks a lot, Dave!!

Ok … I did read, as promised … and found your description, using simple terms that I could understand, perfectly clear – and even enjoyable. I especially appreciated the derivation of the sin wave using logic, rather than math. I believe a point you made toward the end is very good. Many times the math simply get in the way of the understanding of those of us who are less mathematically inclined. It is unfortunate that this mental block is going to prevent me from understanding much of anything else you have to say on the matter of Quantum mechanics. Is there any way you can ‘describe’ your way through that discussion? Anyway, I’m not clear on your motivation. Are you interested in describing to us how the laws are derived? Or, are you more interested in considering the implications of such findings. If the former – I’m lost. If the latter – I’m ready. In any case … keep the posts on Quantum theory coming … I’ll do my best. D

Thanks, Dave, for taking the time to read the old post and commenting – much appreciated!!

It is interesting that you say you ‘appreciated the derivation of the sin wave using logic, rather than math’, because actually this sort of thinking, in my point of view, _is_ the essence of math (and the way math is used in theoretical physics in particular). If you can understand it this way, following some rules and routines to crunch out solutions is easy. It’s rote.

This is what Feynman was so extraordinary gifted at – explaining the gist of ‘math’ though to speak, in a precise way, without writing a single formula. His lectures on Quantum Electrodynamics are masterpieces in that respect: http://www.vega.org.uk/video/subseries/8

Thanks for asking about my motivation – I have actually started crafting a post about my motivation today :-D. It is difficult to respond in a short comment, but I think it is as follows: I had always enjoyed throwing in some sort of ‘science communication’ to my consulting jobs – that holds for IT security as well as renewable energy. Fortunately, there are customers who are interested in how stuff really works – not only for the sake of applying that knowledge, but also for the geeky inspiration.

Trying to explain QFT is my attempt of rising to the ultimate challenge ‘just for the fun of it’ because it is hard – there are no obvious applications in everyday life. I am trying to paint a big picture, trying to explain how physics is used to make sense of the world at large – as in QFT any subfield of physics is needed and connected with each other. I will go easy on my readers next time when I will talk about fields – the concepts used can be illustrated using down-to-earth example.

Again – thanks for your comments!

Looking forward to what comes next! Your student – D

A couple of months ago I bought “Quantum Field Theory Demystified” by David McMahon in Blackwells in Oxford and bumped into my old physics tutor from 40+ years ago. Nice to know he is fit and well. The book has flaws with some arm waving explanations and gaps but as one who hasn’t looked at quantum physics for so long, it is OK. Other books were too compressed for me to follow but this has lots of worked examples. It does get to cover the Feynman Rules and even tries to explain the Higgs mechanism. See Amazon for reviews. I may re-read it when I have finished re-reading Popper’s Conjectures and Refutations and written an entry in my blog looking at criticisms of Popper regarding refutations of refutations. I must confess that it seems to take me 3-4 weeks to write an entry so congratulations Elke on managing a much better rate.

Thanks, Simon – I will consider that book! I have started reading Zee’s QFT in a Nutshell, and as a reviewer on amazon.com says it is not an easy read and McMahon covers the gap.

I am also recapturing the version of QFT I learned many years ago, but as I said in my previous post, the condensed matter / many particle perspective made it actually more difficult for me to recognize that this is really the same theory that should finally help me understand the Higgs field.

This book http://www.quantumfieldtheory.info/ seems very helpful as well – you can read the first chapters online.

As for my publication schedule: I think the secret is simply not to aim at a truly substantial post every time. It does not take that long to craft my quaterly search term poems for example ;-)

I’m looking quite forward to the rest of this series. Thank you.

In QED (I’m pretty sure it was) Feynman managed a pretty awesome trick of hiding the important calculus stuff by way of talking about imagining little clock hands rotating with the length of a path taken, and that’s always impressed me as maybe the greatest feat of science popularization: the variational integrals he’s talking about are … well, clear enough, at least … introduced that way if you know enough calculus to do it, and if you don’t, it’s still a nice easy to visualize scheme that could be used to do a cartoon of the actual calculation.

Thanks, Joseph!

Yes – Feynman starts from a very hands-on proposition about the amplitudes of quantum wave functions, just stating: These are the rules the universe follows, like it or not. He even does this in Vol. 3 of his Physics Lectures – a very unusual approach in a ‘QM beginner’s text book’. Usually educators start with some pseudo-hydrodynamical ‘explanation’, ‘motivation’ of Schrödinger’s equation. Feynman starts with path integrals without using big buzz words.

I have not read QED, but I have seen the related series of video lectures. In the first lecture he also started with explaining path integrals in this way. If I recall correctly he even used geometrical optics and reflection at a mirror as an example / analogy (similar to his beach analogy).

Unfortunately Vol. 2 of his Lectures is not available online yet: This book has a chapter on Feynman’s take on the Principle of Least Action – a verbatim transcript of a lecture (actually unrelated to the rest of the book which is about electrodynamics).

I am most intrigued by the relation between the Principle of Least Action and path integrals (…. the action becoming promoted to the phase of a quantum probability wave in the integrand…), but frankly I have no idea yet how to popularize that. In every book / lecture on QFT the introduction of path integrals in all their glory is considered one of the most difficult conceptual and mathematical leaps a graduate student in physics has to take. There is also a mathematically less daunting route to QFT called the canonical quantization… this is more what I am heading at now… starting from ‘action’ and moving on to its related counterpart in the world of fields. But path integrals are more useful in calculating the real useful stuff so something you have to switch to these concepts. I guess Feynman had always started with path integrals in any lecture.

I find this very difficult to understand with natural language, in my case, English. But I look forward to more because I want the wonderful experience of understanding, and I’ll bet beautiful implications are ahead.

Actually it is hard to resist using math :-) This is sort of an experiment for me!

Hi Elke, I definitely need to brush up on my quantum mechanics ;-) Yet what I understand from your thoughts fits into my mental model of interconnected systems influencing each other. BR, Lucas

Thanks, Lucas! I try to make this series of posts rather self-consistent to that it not required to do additional research. Admittedly, I am not sure how far I will get without quoting math. Currently I try to apply a pop-sci approach following the advice (if I recall correctly: by Stephen Hawking) not to add a single equation as any equation in a popular science book will cut the number of readers in half.

QFT is even more ‘mathy’ than ordinary quantum mechanics – I think I will not be able to avoid math altogether. I hope that there are some math geeks out there (?!)

Hi Elke–that’s the nicest summary of the boundary between classical and quantum mechanics I have seen in a long while. Maybe even ever :-)

As a reward you are now allowed to watch this wonderful talk about one of my physics “Heros” Paul Dirac. :-) While others–Schrödinger for example–are often mentioned for their contributions to the start of QM I think Dirac may have been much more fundamental in its early years. Here’s the link. It’s a great talk.

http://ww3.tvo.org/video/173055/graham-farmelo-paul-dirac-and-mathematical-beauty

Thanks, Maurice! I think i have seen this talk – or a similar one by Graham Farmelo – when I read Farmelo’s biography on Dirac. Yes, in pop-sci history of physics Dirac seems often ousted by Bohr, Heisenberg and Feynman, but finally the introverted geeks do win :-) I feel Farmelo’s biography added to his popularity.

I was also very impressed by the fact that Dirac was trained as an engineer. There is maybe some resemblence to young Newton and young Feynman who built stuff and made experiments as children. (BTW I recommend the biographies of both by James Gleick)