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Path integrals, Feynman path integrals, and functional integrals are synonyms

Generic quantum physicists won't reserve "Feynman integrals" for a finite-dimensional integral

Sometimes, one must write things because someone is wrong on the Internet. And sometimes, even physics bloggers seem to be excited enough to publish an apparently wrong blog post whose only purpose seems to be to promote the opposite of an important truth.

Almost exactly one year ago, Jacques Distler published some bizarre comments claiming that quantum mechanics of relativistic particles is consistent even without the introduction of antiparticles and/or multiparticle states – i.e. without quantum fields.

Well, they're not. Quantum fields – and therefore antiparticles (or creation considered along with annihilation) – are an unavoidable consequence of quantum mechanics combined with special relativity. If you try to define any theory that evolves one particle's wave function, it's unavoidable that it has a positive probability to spread superluminally. To stay inside the light cone, you simply need both contributions with positive and negative frequencies: you need to create particles and annihilate antiparticles, too.

Today, Tetragraviton posted a bizarre text claiming that path integrals and contour integrals are "totally different" things.

Well, he's just heavily exaggerating, to say the least. First, let me mention that the phrase "path integral" is sometimes used for line integrals (in some real space) or contour integration (in the complex plane). These concepts are clearly simpler and Tetragraviton didn't mean those.

He did mean some "path integrals" that aren't just line or contour integrals. But as long as it is the case, he must have meant functional integrals or Feynman path integrals. And as general concepts, these three terms are exactly equivalent. The name "Feynman" is associated with the path integrals or functional integrals because Richard Feynman has used those infinite-dimensional integrals to define his path integral formulation of quantum mechanics.

The path integral formulation is the quantum mechanics' exploitation of the concept of the action – and therefore the Lagrangian (the action is the time integral of the Lagrangian, or spacetime integral of the Lagrangian density). Instead of saying\[

\delta S = 0

\] as in classical physics which says that only the trajectory with the extremized action is allowed, quantum mechanics says that the probability amplitude of any evolution is nonzero and given by \[

{\mathcal A}(x_i\to x_f) = \int {\mathcal D}x(t) \exp(iS[x(t)]/\hbar)

\] the infinite-dimensional functional integral over all trajectories that start at \(x_i\) and end at \(x_f\). The integrand is the exponential of the action multiplied by the imaginary unit – so the absolute value of the integrand is the same for all trajectories. The phase is chaotically changing almost everywhere so the integrand tends to cancel itself. The vicinity of the histories that obey \(\delta S = 0\) are exceptions. The phase is nearly stabilized there, the nearby trajectories constructively interfere, and it's an explanation using the path integral approach to quantum mechanics why classical physics emerges as a limit.

The integrals over \({\mathcal D} x(t)\) may be considered the path integral treatment of quantum mechanics of one particle – or, equivalently, a 0+1-dimensional field theory. Fields in \(d+1\)-dimensional field theory depend on \(d\) spatial coordinates and one time. If \(d=0\), there are not spatial coordinates and the "fields", often denoted \(x\), only depend on time, just like \(x(t)\).

In 1933 when Feynman was 15, Dirac already had some vague observations that the path integral is an equivalent way to calculate the amplitudes in quantum mechanics. Feynman made all these things explicit. He also extended the method to field theory which is straightforward. In the \(d+1\)-dimensional field theory, we are integrating\[

{\mathcal A}(i\to f) = \int {\mathcal D}\phi(x,y,z,t) \exp(iS[\phi(t)]/\hbar).

\] More work has to be done to deal with the initial conditions encoding the initial and final state etc. But the "bulk" of the integral only differs from the previous one by renaming \(x\) to \(\phi(x,y,z)\) with the added dependence on the spatial coordinates. The methodology is the same – and so is the proof that this calculation of the amplitudes is equivalent to the Heisenberg or Schrödinger pictures.

Both the integrals over \(\int {\mathcal D}\phi(x,y,z,t)\) as well as over \(\int{\mathcal D}x(t)\) may be chosen as the starting point to derive the Feynman diagrams in quantum field theories. The propagators arise from some two-point function of fields in the first integral over quantum fields; and as the integrals over trajectories of one particle in the second integral over one-particle histories. The vertices arise from cubic and higher-order terms in the Lagrangian in the approach using quantum fields; and from the added possibility for particles to split or merge in the case of mechanics.

When we talk about the "Feynman path integrals" and deliberately make the phrase redundant in this way, we may mean path integrals that integrate over the coordinate space – and not momentum space or phase space. But this may be called a bias of some nitpicky people; there exists no authoritative regulation and no justifiable reasons why "path integrals", "Feynman integrals", "Feynman path integrals", and "functional integrals" shouldn't be considered synonymous by physicists.

So there were two possible reasons why Tetragraviton would write the blog post whose only goal is to obsessively negate the previous sentence. Either he is really confused about something elementary about path integrals; or he is nitpicky in his own way. I decided that the second answer is correct. He (and indeed, he's not the only one) wants to reserve the "Feynman integrals" for integrals used to evaluate the Feynman diagrams with loops that use the Feynman parametrization. Or for some previous "stage" of the calculation of the amplitudes – before we actually introduce the Feynman parameters. It's still the same amplitude we're calculating and I don't think it's a good work with the language to invent names just for "stages" of a calculation, as opposed to well-defined players or tricks.

Well, as you can check by clicking at the previous sentences, the "Feynman parametrization" is the standard name for that trick – "Feynman parameters" and "Feynman trick" represent the same calculations (or the new integration variables in them). On the other hand, there is a very good reason why Feynman integral is redirected to the entry about the path integral formulation. The combination of Feynman and integrals simply looks way too powerful and one doesn't want to reserve this phrase for some finite-dimensional integral of a special kind.

So even though I know that people in the "amplitude business" love to use "Feynman integrals" as some finite-dimensional integrals used to evaluate Feynman diagrams with loops, and even Nima and others use the phrase in this way, I find his efforts to disentangle the term "Feynman integral" from the general functional integrals or the path integral formulation of quantum mechanics to be a lost cause. The rule that "Feynman integrals" represents particular finite-dimensional integrals should be considered a jargon of the "amplitude subcommunity" of theoretical physics community, not a rule adopted by all quantum physicists.

I don't know why people sometimes get so obsessed about their terminological idiosyncrasies. Even if we decided that his usage of the "Feynman integral" for the finite-dimensional integral is "more correct", it won't change anything about the fact that it's just some words and lots of smart people and brilliant papers have used "Feynman integrals" for the path integrals. For example, take this 1969 paper by L.D. Faddeev (from the Faddeev-Popov ghosts) that has almost 900 citations and did use the term "Feynman integral" for the path integrals. It's the first paper you get if you search for "Feynman integral" at Google Scholar.

Yes, it's almost half a century ago and newer papers generally try to avoid using the "Feynman integral" for infinite-dimensional path integrals. But the identification is just way too tempting and even if you managed to "ban it" in the amplitude industry, you won't ban it in the whole physics community. In particular, when general relativists try to play with quantization, they almost universally use "Feynman integral" for the path integrals. That includes a 1957 book by Misner but also more recent papers by Rovelli, if I have to add a provocative name.

Scholarpedia and Encyclopedia of Mathematics also agree that the term "Feynman integral" is sometimes used for path integrals.

And by the way, I am convinced that Feynman would agree with me that the "Feynman integral" shouldn't be reserved for some particular finite-dimensional integral evaluating a Feynman integral. That usage looks like an excessive focus on words and details about them, something Feynman always opposed. "Feynman integral" sounds like a simple concept and it should represent the big idea, not a special small one.

Well, there's an actual fun story about that. At a 1979 conference, someone has used the term "Feynman integral" (probably in the finite-dimensional sense) and Feynman asked: "What is the Feynman integral?"

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