# Where does Field Theory come from?

About my background: I'm currently a 4th year undergrad, and planning to do a PhD in theoretical physics. I think I have a decent understanding of basic physics, and I know how to do calculations in quantum field theory (e.g. calculate scattering amplitudes, renormalization), but I still feel sketchy on some basic concepts.

The question I have is where does the idea of "field theory" come from?

Here is my understanding: A "field" is something that's defined at every point in space and time. Some fields have a "field theory" attached. For example, the temperature in a room is a field, but there isn't any field theory other than perhaps the fluid dynamics of the air.

In field theory, first we have a Lagrangian (or Hamiltonian, or we have an Action that we use to get the Lagrangian). Then we use the principle of least action/Euler-Lagrange equations to derive a set of equations that the fields obey. In classical field theory the fields are classical physical quantities; in quantum field theory the fields are quantum operators. Classical electromagnetism [gravitation] is a classical field theory in the sense that we can write down a Lagrangian and follow the field theory recipe to derive Maxwell's equations [Einstein equations].

Maybe it's just me, but the concept of "field theory" seems somewhat pulled out of a hat. Why is field theory so important and widely applicable? Is the Euler-Lagrange equation/least action principle a fundamental law of physics?

There are many articles/lectures/books on "where does QFT come from?" They usually explain how in order to have relativistic quantum mechanics we need to have a theory of not particles, but fields. Or, how one can describe condensed matter systems in terms of emergent QFTs. But these explanations only explain why theories of certain physical systems turn out to be field theories, not how the field theory concept (both classical and quantum) exists in the first place.

Edit/clarification: I think another way to phrase my question is how does one go from classical mechanics to field theory? In mechanics, the idea of Lagrangian/least action is clear since it's equivalent to Newton's laws. But it is not obvious why it should be possible to export this formalism to other areas of physics with no direct connection to classical mechanics.

• The canonical reference is Weinberg's The quantum theory of fields, Vol. 1. – AccidentalFourierTransform Dec 7 '18 at 22:20
• A field theory just describes how the values of the field change in time (also called the dynamics). Your example about temperature does have an associated field theory. The dynamics of the field can be classical or quantum in modern physics, @AccidentalFourierTransform has given the reference for the quantum theory of fields. – Bobak Hashemi Dec 7 '18 at 22:21
• @HalberdRejoyceth Sure, I will summarise the content of a four-hundred-pages book in this comment box. I might need some help from my good ol' friend Fermat. Here it goes (1/500). – AccidentalFourierTransform Dec 7 '18 at 22:38
• @aqualone then your question is "does every field theory have a lagrangian description", to which the answer is no. If you are asking which field theories have been lagrangian descriptions, or why are lagrangians important, that's an interesting question which I never have found a satisfactory answer to... Vladimir Arnolds book on classical mechanics seemed to be moving in the right direction, but I never got a through it – Bobak Hashemi Dec 7 '18 at 23:22
• Related: physics.stackexchange.com/q/13157/2451 and links therein. – Qmechanic Dec 8 '18 at 19:08

I'm going to try to tackle this one piece-by-piece, because I think the questions you are asking are important and, without outside help, one can spend years extremely confused about where the basic ideas of field theory come from.

(Warning: I got a little carried away and this became much longer than I expected.)

• "Where does the idea of "field theory" come from?"

Field theories come from the very simple fact that we observe fields in nature. We see electromagnetic fields exist in response to charged particles and currents. We see velocity fields in fluid flow. We see fields describing temperature and mass density. Some of these fields (such as the fluid flow, temperature, and mass density) seem to lose their "field-like" properties at small scales (i.e. water is a collection of discrete molecules), but the electromagnetic and gravitational fields seem to be truly field-like (i.e. they are described by a continuous function everywhere in space). Thus, in order to describe nature, we need to be able to describe fields.

• "Why is field theory so important and widely applicable?"

Partly because we observe classical fields in nature, and partly because field theories that are Lorentz invariant, upon quantization, immediately give us the particles we see in nature (more on that later). However, the true power of field theory is the ability to describe any system that can be approximated by a continuous medium.

• "Is the Euler-Lagrange equation/least action principle a fundamental law of physics?"

As far as classical physics is concerned, I would definitely say yes. There isn't a (classical) theory I know of that can't be described (in some way) as a principle of least-action. If this feels ad-hoc, consider the fact that classical theories (of either particles or fields) obey equations of motion. Particles obey Newton's laws, electromagnetic fields obey the Maxwell equations, etc. These equations of motion can all be formulated in a least-action way. For many physical systems, there is more than one action that leads to the same equations of motion, so how "fundamental" a given action is is a little ambiguous, but the fact that you can derive the equations of motion for a given theory from an action principle seems to suggest that the Euler-Lagrange equations are extremely convenient and useful, if not fundamental.

Hamiltonians seem to be fundamental as well, as the Hamiltonian for a given classical path is something we can measure (namely, its energy).

• "But these explanations only explain why theories of certain physical systems turn out to be field theories, not how the field theory concept (both classical and quantum) exists in the first place."

The classical field theory concept, again, comes from the fact that we observe fields in nature, and thus we require a method of describing them.

The quantum field theory concept is a little trickier, but consider the following short thought experiment to convince yourself that we need it: An electron has been prepared in a state $$|x_0\rangle+|x_1\rangle$$. What is the electric field associated with this electron? If the electron were at position $$x_0$$, it would produce an electric field $$E_{x_0}(x)$$, and if the electron were at position $$x_1$$ it would produce an electric field $$E_{x_1}(x)$$. However, the electron is in a superposition of both of these positions. If you could measure the electric field, you could determine the electron's position, and so the electric field and the electron position must be entangled. In particular, the full state of the system (electron + electric field) must take the schematic form

$$|\psi\rangle=|x_0\rangle\otimes|E_{x_0}\rangle+|x_1\rangle\otimes|E_{x_1}\rangle,$$

(up to a relative phase between the two states). This suggests that, since electrons are quantum-mechanical particles and they are sources for electric fields, electric fields themselves must be quantized. The same reasoning could follow for any field the electron produces (magnetic, gravitational, etc.). This little thought experiment tells us that fields are quantized.

• "...how does one go from classical mechanics to field theory?"

In exactly the same way that one goes from classical mechanics to standard quantum mechanics! We identify a Hamiltonian and define a wavefunction. In particular, given a field variable $$\phi(\textbf{x})$$, you define a "wave-functional" $$\Psi[\phi_0]$$ which, in a sense, tells you the probability amplitude for measuring your system to have the field configuration $$\phi_0$$, and obeys a Schrodinger equation

$$\hat{H}\Psi[\phi]=i\hbar\frac{\partial}{\partial t}\Psi[\phi],$$

where the Hamiltonian has to be specified by your particular theory.

Usually one ditches the wave-functional approach and just defines states $$|\Psi\rangle$$ with $$\langle\phi_0|\Psi\rangle=\Psi[\phi_0]$$, where $$|\phi_0\rangle$$ is a state of definite field-configuration $$\phi_0(\textbf{x})$$. The field variable itself is then promoted to an operator $$\hat{\phi}$$, for which $$|\phi_0\rangle$$ are its eigenstates. After this, you can find the spectrum of your Hamiltonian and everyone is happy.

(Next is my own question, but one I'm sure you're asking too.)

• "So what the hell does all this have to do with particles and relativity?"

Fair question. It turns out when you perform the above procedure on electromagnetism, the spectrum of the resulting theory (QED) looks an awful lot like it contains non-interacting particles of definite momentum and spin, and whose momentum $$\textbf{p}$$ and energy $$E$$ satisfy $$E^2-\textbf{p}^2c^2=0$$. If the states in this theory are to describe particles, then they must be massless (since $$E^2-\textbf{p}^2c^2=m^2c^4$$ in relativity), and thus must travel at the speed of light. Furthermore, the energy of such a state can be shown to be $$E=\hbar\omega$$, where $$\omega$$ is the frequency of the corresponding classical plane-wave. This exactly corresponds with the results found in the photoelectric effect and the black-body spectrum. In essence, as soon as you try to quantize the electromagnetic field, photons essentially emerge (almost magically).

After you can accept that photons come from the quantization of the electromagnetic field, it shouldn't be too difficult to consider that perhaps the other particles in nature can be described by fields too. Indeed, the Dirac field (which has no classical analogue, and so is a little difficult to intuit) describes all known Fermions, the non-Abelian gauge fields (a certain generalization of the electromagnetic fields) describe gluons, $$W$$ Bosons, and $$Z$$ Bosons, and scalar fields (which are the simplest possible fields one could consider) describe the Higgs Boson.

In conclusion:

One could argue that field theory comes from the need to describe relativistic particles in quantum mechanics, and that is one way to get to the conclusion that the universe should consist of quantum fields. Another way would be to simply realize that the world consists of classical fields, and one necessarily needs a quantum description of them.

I hope this helps you understand at least the basic notion of where quantum field theory comes from and why it is so powerful.

PS: The reason you feel sketchy about the fundamental concepts of field theory is that the standard curriculum (based on books like Peskin and Schroeder) doesn't teach field theory. It more or less provides a crash course in the narrow topic of calculating scattering amplitudes from some weakly coupled field theory models. Understanding field theory goes far beyond calculating Feynman diagrams. Field theory is a much richer and broader field than textbooks would have you believe.

Classical fields are all over the place in describing "big" (extended) systems with infinite degrees of freedom, and their long-distance interactions and medium undulations. Think of water waves (the water interface is a field), or electromagnetism, or the air density field that conducts sound.

Possibly the best pathway to "internalize" quantization of fields is the quantization of lattice vibrations into phonons. You start with a classical field which varies and oscillates in space, like deformation fields on a lattice, or electromagnetic fields in vacuum, etc... You study the classical oscillations, waves, etc in such. The central takeaway is that such oscillations, elastic jiggling/"sound" in lattices, EM waves, etc... can be described classically by an infinity of suitably packaged harmonic oscillators ("normal modes of oscillation"). Classically. (Of course, the obvious efficient language for that collective description is the Lagrangian and Hamiltonian formalism.)

Quantizing oscillators is straightforward, especially in Dirac's creation/annihilation operator language. In the late 20s, Jordan and collaborators (and Dirac by himself) devised an elegant language to do that, and quantized fields by suitably organizing the collective quantization of this infinity of oscillators, mindful of the structure and the symmetries of the theory ("field quantization"). The space of this infinite collection of oscillators is called "Fock space", and serves to systematically describe phonons for the elasticity waves, photons for electromagnetism, fermions for other types of fields, etc. All particles and their interactions, created, destroyed, and morphing, are thus described by quantum fields.

• different topic.Can you look at the comment in my answer here : physics.stackexchange.com/questions/445774/… . this paper mcgreevy.physics.ucsd.edu/s13/final-papers/… which says that the pertubation series is divergent, even after renormalization. I am really puzzled. – anna v Dec 8 '18 at 7:21
• Oh, a long and messy story... indeed, the perturbation series is asymptotic, and not convergent ... but as such, in QED it works asymptotically quite well... which that nice summary of the familiar pathologies fails to explain... Asymptotic series are difficult to understand, but nobody should discourage one from calculating the 11th order in QED because it would not improve on the 10th at hand. (The divergences handled by renormalization are such due to infinite cutoffs, not infinite terms in the series.) – Cosmas Zachos Dec 8 '18 at 15:09
• It should also be mentioned that, practically, the divergence of the QED series probably isn’t a problem, as there exist analytic techniques (particularly Padé resummation) which improve on the convergence of analytic series. Whether this works for QED is yet to be shown, but I’m optimistic. – Bob Knighton May 9 at 11:19

The question I have is where does the idea of "field theory" come from? ... Maybe it's just me, but the idea of a field theory just seems pulled out of a hat.

The notion of a field, was not 'pulled out of a hat'. As explained by Maxwell, it came from Faraday. In 1876 Maxwell published a paper in Philosophy Magazine titled On Faradays Lines of Force:

It is by the use of analogies of this kind that I have attempted to bring before the mind, in a conveniant and manageable form, those mathematical ideas which are neccessary to the study of the phenomena of electricity. The methods are generally those suggested by the process of reasoning which are found in the researches of Faraday [to which he refers to series xxxviii of Faradays Experimental Researches and Philosophy Magazine 1852] and which, though they have been interpreted mathematically by Prof. Thomson and others, are generally supposed to be on an indefnite and unmathematical character, when compared to those employed by professed mathematicians ...

The limit of my design is to shew how, by a strict application of the ideas and methods of Faraday, the connexion of the very different orders of phenomena which he has discovered may be placed before the mathematical mind ... I have in the first place, to explain and illustrate the idea of 'lines of force.'...

The notion of 'lines of force' answers a question that was raised by Newton who struggled to answer it: how can force act at a distance?

• I think you'll find it's 'Philosophical Magazine' rather than 'Philosophy Magazine' e.g. "LVIII. On the physical character of the lines of magnetic force Michael Faraday Esq. D.C.L. F.R.S.Philosophical Magazine Series 4 Volume 3, 1852 - Issue 20", via tandfonline.com/doi/abs/10.1080/14786431003659180 (I tried to edit the Answer above, but it won't accept an edit of less than 6 characters, even if I retype the whole word. – iSeeker Dec 11 '18 at 18:07

Let me sketch a path from actions to QFT, where the F part comes up before the Q part, but at least the Q part makes sense of why we expect stationary actions in the first place:

• Before either Einstein or QM had contradicted Newton, we had a classical mechanics viz. $$S=\int dt L(q,\,dq/dt)$$. Here $$q$$ is a coordinate dependent on a parameter $$t$$.
• Unlike Galilean relativity, special relativity blends space and time, and requires us to make all spacetime coordinates parameters. So now we have something more like $$S=\int d^n x \mathcal{L}(\phi,\,\partial_\mu\phi)$$ in an $$n$$-dimensional spacetime. There have been some notational changes, such as the coordinate now being $$\phi$$ instead of $$q$$. At this point, we call it a field.
• Did field theory predate SR? Sure; for example, Maxwell's equations feature fields, and the equations themselves are derivable from a Lagrangian. But while it's historically true that we got to SR from there, it's also true that if you take Lorentz invariance as your starting point, and try to promote a $$U(1)$$ invariance from global to local, the Maxwell Lagrangian comes up organically. So if we switch from being historians to being physicists, we can say field theory is just a tweak to Lagrangian mechanics we need to move beyond Galileo.
• In QM, the time parameter is not an observable; instead, observables depend on it. Similarly, in QFT fields are observables that depend on the parameters $$x^\mu$$, none of which are observables. This is not a weird repudiation of what QM teaches you earlier in your undergraduate days. It's important to see QFT as an update to a classical field theory, by the same technique as we get QM from its classical counterpart, instead of seeing QFT as a repaired version of QM.
• Mind you, it's also important to see QFT as inevitable when you try to simultaneously concede quantum and relativistic effects. For example, it's easy to show non-relativistic QM's usual probability interpretation breaks down, at which point you realise $$\psi$$ will have to be a field from which we can obtain positive or negative particle numbers. If I were giving a course of lectures on QFT, I think I'd actually start its motivation there.
• Feynman's path integral formalism motivates a stationary action.
• Where does any physical theory come from? We notice something in need of explanation, an attempt at it makes predictions, those predictions turn out true, and they retroactively become something that would have similarly required that theoretical innovation if they were already known. For example, QFT explains spin and antimatter. We already knew the former, but it ended up predicting the latter.
• The specific QFTs we use in, say, the Standard Model also make sense of more specific empirical truths (be they things that were already known or predicted), such as kaon oscillation, neutrino oscillation or CP violation.
• You can read any and all answers posted here and every textbook ever written, but you might still come away thinking, "So, they use these things to justify that thing, but it also double backs on itself". This is not circularity; it's just that "A is fundamental while B is a consequence of A" is often a silly statement, because the explanatory reasoning can go in multiple directions. Do we get non-Galilean relativity because of electromagnetism, or vice versa? You don't need to answer that to do physics; you just need to see how later theories expand what goes on in the earlier ones. As I said, field theory is what happens to classical mechanics when you link time to space. Why is it linked to space? In theory you could have a no-field universe, but ultimately the data says no to that.