Path Integrals, Feynman Path Integrals, In Addition To Functional Integrals Are Synonyms

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

Sometimes, ane must write things because somebody is incorrect on the Internet. And sometimes, fifty-fifty physics bloggers seem to endure excited plenty to issue an plainly incorrect weblog post whose exclusively purpose seems to endure to promote the contrary of an of import truth.

Almost precisely ane twelvemonth ago, Jacques Distler published unopen to bizarre comments claiming that quantum mechanics of relativistic particles is consistent fifty-fifty without the introduction of antiparticles and/or multiparticle states – i.e. without quantum fields.

Well, they're not. Quantum fields – as well as thus antiparticles (or creation considered along amongst annihilation) – are an unavoidable resultant of quantum mechanics combined amongst special relativity. If you lot test to define whatever theory that evolves ane particle's moving ridge function, it's unavoidable that it has a positive probability to spread superluminally. To remain within the low-cal cone, you lot just demand both contributions amongst positive as well as negative frequencies: you lot demand to practise particles as well as annihilate antiparticles, too.

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




Well, he's just heavily exaggerating, to say the least. First, allow me advert that the phrase "path integral" is sometimes used for line integrals (in unopen to existent space) or contour integration (in the complex plane). These concepts are clearly simpler as well as Tetragraviton didn't hateful those.




He did hateful unopen to "path integrals" that aren't just line or contour integrals. But every bit long every bit it is the case, he must own got meant functional integrals or Feynman path integrals. And every bit full general concepts, these iii damage are precisely equivalent. The holler "Feynman" is associated amongst 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 activity – as well as thus the Lagrangian (the activity is the fourth dimension integral of the Lagrangian, or spacetime integral of the Lagrangian density). Instead of saying\[

\delta south = 0

\] every bit inward classical physics which says that exclusively the trajectory amongst the extremized activity is allowed, quantum mechanics says that the probability aAmplitude of whatever development is nonzero as well as given past times \[

{\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 \(x_i\) as well as halt at \(x_f\). The integrand is the exponential of the activity multiplied past times the imaginary unit of measurement – as well as so the absolute value of the integrand is the same for all trajectories. The stage is chaotically changing almost everywhere as well as so the integrand tends to cancel itself. The vicinity of the histories that obey \(\delta south = 0\) are exceptions. The stage is nearly stabilized there, the nearby trajectories constructively interfere, as well as it's an explanation using the path integral approach to quantum mechanics why classical physics emerges every bit a limit.

The integrals over \({\mathcal D} x(t)\) may endure considered the path integral handling of quantum mechanics of ane particle – or, equivalently, a 0+1-dimensional land theory. Fields inward \(d+1\)-dimensional land theory depend on \(d\) spatial coordinates as well as ane time. If \(d=0\), at that spot are non spatial coordinates as well as the "fields", oft denoted \(x\), exclusively depend on time, just similar \(x(t)\).

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

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

\] More piece of work has to endure done to bargain amongst the initial atmospheric condition encoding the initial as well as in conclusion nation etc. But the "bulk" of the integral exclusively differs from the previous ane past times renaming \(x\) to \(\phi(x,y,z)\) amongst the added dependence on the spatial coordinates. The methodology is the same – as well as and so is the proof that this calculation of the amplitudes is equivalent to the Heisenberg or Schrödinger pictures.

Both the integrals over \(\int {\mathcal D}\phi(x,y,z,t)\) every bit good every bit over \(\int{\mathcal D}x(t)\) may endure chosen every bit the starting indicate to derive the Feynman diagrams inward quantum land theories. The propagators arise from unopen to two-point business office of fields inward the kickoff integral over quantum fields; as well as every bit the integrals over trajectories of ane particle inward the 2d integral over one-particle histories. The vertices arise from cubic as well as higher-order damage inward the Lagrangian inward the approach using quantum fields; as well as from the added possibility for particles to split upwardly or merge inward the instance of mechanics.

When nosotros beak close the "Feynman path integrals" as well as deliberately brand the phrase redundant inward this way, nosotros may hateful path integrals that integrate over the coordinate infinite – as well as non momentum infinite or stage space. But this may endure called a bias of unopen to nitpicky people; at that spot exists no authoritative rule as well as no justifiable reasons why "path integrals", "Feynman integrals", "Feynman path integrals", as well as "functional integrals" shouldn't endure considered synonymous past times physicists.

So at that spot were 2 possible reasons why Tetragraviton would write the weblog post whose exclusively destination is to obsessively negate the previous sentence. Either he is genuinely confused close something simple close path integrals; or he is nitpicky inward his ain way. I decided that the 2d respond is correct. He (and indeed, he's non the exclusively one) wants to reserve the "Feynman integrals" for integrals used to evaluate the Feynman diagrams amongst loops that work the Feynman parametrization. Or for unopen to previous "stage" of the calculation of the amplitudes – earlier nosotros genuinely innovate the Feynman parameters. It's withal the same aAmplitude we're calculating as well as I don't recollect it's a expert piece of work amongst the linguistic communication to invent names just for "stages" of a calculation, every bit opposed to well-defined players or tricks.

Well, every bit you lot tin cheque past times clicking at the previous sentences, the "Feynman parametrization" is the criterion quest that line a fast ane on – "Feynman parameters" as well as "Feynman trick" stand upwardly for the same calculations (or the novel integration variables inward them). On the other hand, at that spot is a rattling expert argue why Feynman integral is redirected to the entry close the path integral formulation. The combination of Feynman as well as integrals just looks manner every bit good powerful as well as ane doesn't desire to reserve this phrase for unopen to finite-dimensional integral of a special kind.

So fifty-fifty though I know that people inward the "amplitude business" dearest to work "Feynman integrals" every bit unopen to finite-dimensional integrals used to evaluate Feynman diagrams amongst loops, as well as fifty-fifty Nima as well as others work the phrase inward this way, I detect his efforts to disentangle the term "Feynman integral" from the full general functional integrals or the path integral formulation of quantum mechanics to endure a lost cause. The dominion that "Feynman integrals" represents especial finite-dimensional integrals should endure considered a jargon of the "amplitude subcommunity" of theoretical physics community, non a dominion adopted past times all quantum physicists.

I don't know why people sometimes larn as well as so obsessed close their terminological idiosyncrasies. Even if nosotros decided that his usage of the "Feynman integral" for the finite-dimensional integral is "more correct", it won't alter anything close the fact that it's just unopen to words as well as lots of smart people as well as bright papers own got used "Feynman integrals" for the path integrals. For example, accept this 1969 newspaper past times L.D. Faddeev (from the Faddeev-Popov ghosts) that has almost 900 citations as well as did work the term "Feynman integral" for the path integrals. It's the kickoff newspaper you lot larn if you lot search for "Feynman integral" at Google Scholar.

Yes, it's almost one-half a century agone as well as newer papers mostly test to avoid using the "Feynman integral" for infinite-dimensional path integrals. But the identification is just manner every bit good tempting as well as fifty-fifty if you lot managed to "ban it" inward the aAmplitude industry, you lot won't ban it inward the whole physics community. In particular, when full general relativists test to play amongst quantization, they almost universally work "Feynman integral" for the path integrals. That includes a 1957 volume past times Misner but also to a greater extent than recent papers past times Rovelli, if I own got to add together a provocative name.

Scholarpedia as well as Encyclopedia of Mathematics also concord that the term "Feynman integral" is sometimes used for path integrals.

And past times the way, I am convinced that Feynman would concord amongst me that the "Feynman integral" shouldn't endure reserved for unopen to especial finite-dimensional integral evaluating a Feynman integral. That usage looks similar an excessive focus on words as well as details close them, something Feynman ever opposed. "Feynman integral" sounds similar a simple concept as well as it should stand upwardly for the big idea, non a special modest one.

Well, there's an actual fun flush close that. At a 1979 conference, somebody has used the term "Feynman integral" (probably inward the finite-dimensional sense) as well as Feynman asked: "What is the Feynman integral?"