How Path Integrals Mirror Feynman's Personality Traits

The two-day quiet is to a greater extent than ofttimes than non due to the September 19th wheel trip, 100 kilometers starting inwards the mountains (Bohemian Forest), sorry.

Courage, playfulness, analogies, close upward in addition to calculate (calculation instead of words), lots of calculations extending uncomplicated rules, numbers instead of philosophy, don't surrender easily

The most successful theories of classical physics may hold out formulated inwards price of the regulation of to the lowest degree action. We may consider option histories \(x_i(t)\) where some observables \(x_i\) depend on fourth dimension \(t\). The regulation says that the activeness \(S\) which is a functional of the history (a collection of functions) \(x_i(t)\) is minimized for the history that is really allowed yesteryear the laws of physics:\[

\delta south [x_i(t)] = 0.

\] Paul Dirac has been convinced that this elegant formalism of classical physics – based on the concept of the activeness – should receive got a correspondingly squeamish role inwards quantum mechanics. And he institute a adept guess. In quantum mechanics, ane could mayhap calculate the probability amplitudes for the development of \(x_i(t_1)\) to \(x_i(t_2)\) equally \(\exp(iS/\hbar)\).

That was squeamish in addition to Dirac presented some basic declaration why the Lagrangian (whose integral over fourth dimension gives the action) is related to the Hamiltonian but he didn't exercise much alongside this idea. It looked also heuristic to him.



Richard Feynman did much to a greater extent than than that. He turned the path integrals into a large manufacture inside quantum mechanics, calculated the amplitudes for some basic systems using the path integrals, in addition to derived the Feynman diagrams from the path integrals applied to quantum plain theories. He developed many tricks to really calculate the path integrals – the Feynman parameterization, amidst many other things – in addition to he later on added some clever cherries on a pie such equally the Faddějev-Popov ghosts (Feynman introduced them first, piece quantizing full general relativity using path integrals).

Path integrals became the key engine used yesteryear particle physicists to calculate things inwards quantum plain theory equally good equally string theory.



It's like shooting fish in a barrel to rationalize the history – the hindsight makes it trivial – but I can't resist to say that at that spot seem to hold out really adept reasons why Feynman, in addition to non anyone else (not fifty-fifty Dirac), was able to acquire this far. He had a comparative payoff because the personality traits that are evidently helpful to exercise progress alongside path integrals are highly correlated alongside Feynman's personality traits – mayhap traits that may hold out seen inwards many other places of his intellectual in addition to fifty-fifty everyday life.

Let me explicate this declaration a niggling bit.

Feynman liked analogies – for example, he would explicate mechanics using analogous electrical circuits. He realized that the activeness was a really powerful concept inwards classical physics which is why it seemed intriguing to believe that it could hold out powerful inwards quantum mechanics, too. Dirac has sketched the exclusively plausible way how the activeness could enter. The particles or other physical systems "sniff" all trajectories or histories at once. This is a generalized version of a particle's going through "both slits" inwards Feynman's favorite "template" for all quantum mechanical experiments.

Now, "sniffing all possible [classical] answers" is something that is arguably Feynmanesque yesteryear itself. All possible answers are given a adventure to influence the total number (this positive mental attitude was mayhap linked to his wishing to "look at all things from many perspectives"); in addition to ane must exclusively specify the rules of the contest or cooperation betwixt these answers. Complex amplitudes are everywhere inwards quantum mechanical expressions, nosotros wishing something that depends on the action, in addition to thus at that spot has to hold out an aAmplitude that depends on the action, in addition to it has to hold out \(\exp(iS/\hbar)\). Note that Feynman has also considered \(\exp(i\pi)+1=0\) to hold out the most beautiful equation of mathematics in addition to thus the imaginary exponents were sure as shooting something he liked. When all of state-of-the-art physics mightiness hold out reduced to straightforward calculations of sums of \(\exp(iS/\hbar)\) terms, that sounds pretty amazing in addition to ane should pass some issue energy on it, right?

OK, for these reasons, the path integral for the amplitudes\[

{\mathcal A}_{fi} = \int {\mathcal D}x(t)\,\exp(iS / \hbar)

\] alongside the appropriate initial in addition to concluding weather condition was in all probability to a greater extent than natural in addition to attractive for Feynman than for 95% of theoretical physicists of his time. But Dirac could figure out this basic dominion equally well. Dirac stopped also early piece Feynman continued. Why? Well, when Feynman started to continue, Dirac was roughly forty years old. Maybe it's already also much. Maybe it's non also much.

But Dirac has already had lots of groundbreaking discoveries inwards the mass of his achievements. It's natural to tardily down. Moreover, these achievements receive got made Dirac a crucial theorist. On the other hand, all the extra insights most the path integral that Feynman did were applications of some basic ideas. In this sense, Feynman was an industrialist similar Edison.

He liked to ready radios, play sophisticated games alongside uncomplicated plenty toys. That's different from some other kids in addition to physicists who prefer to play alongside complicated toys.

Now, the path integral \(\int {\mathcal D}x(t)\) is infinite-dimensional – which sounds complicated – but from the beginning, peculiarly if the integrand is familiar enough, it's a "repeated application" of something that is uncomplicated enough, something that Feynman knew really good from the finite-dimensional integrals.

So although the infinite-dimensional graphic symbol looks hard, the path integrals seem to hold out a game that has uncomplicated rules inwards principle. It's similar fixing the radios all the fourth dimension or picking the locks all the time. So the path integrals were just a game that he just had to like. You know, the infinite that is integrated over is flat. It's non curved in addition to it doesn't receive got a terribly complicated topology. It's good. The firstly integrand that nosotros insert – for the harmonic oscillator in addition to the gratuitous plain theory – is a Gaussian business office (times a polynomial prefactor if nosotros demand to acquire some correlation functions). That's straightforward, too. Because it's possible to calculate the finite-dimensional Gaussian integrals, it should hold out possible to calculate the infinite-dimensional ones, too.

Some results for the amplitudes inwards the uncomplicated plenty quantum mechanical systems be inwards the literature, the path integral should hold out formally right, mayhap alongside some modifications, in addition to thus at that spot must be a way to bargain alongside the path integral that produces the right results.

Now, much of Feynman's activity in addition to excitement piece dealing alongside the path integrals was driven yesteryear his "shut upward in addition to calculate" approach to thinking. Lots of people could endeavour to dismiss the path integrals for many reasons – all of which were ultimately proven spurious:

1. The exponential of the imaginary exponent has a constant absolute value – in addition to thus different the Gaussian case, the back upward is non-compact in addition to the integration over "regions at infinity" is jump to attain problems.
2. The functional integral is a continuously infinite dimension in addition to thus it's non fifty-fifty a bound of a finite-dimensional integral.
3. The functional integral may hold out different from the integral when the fourth dimension is sliced, in addition to thus the slicing could neglect to hold out helpful.
4. There is no way to define the infinite-dimensional Riemann integral because there's no way to separate the integration part to little pieces.
5. There is no way to define the Lebesgue integral because the regions of the integration infinite where the integrand has some value are extremely complicated in addition to can't hold out assigned a measure.
6. All the prefactors that appear inwards the partial calculation are singular – infinite or null – inwards the continuum bound which makes the whole calculation indeterminate.
7. Divergent expressions such equally \(1+2+3+4+\dots \) appear fifty-fifty inwards the factors that are sure as shooting important, in addition to those brand finite results impossible.
8. Other generically (UV) divergent integrals (over momenta) appear inwards the loop integrals in addition to they brand finite answers impossible.
9. Long-distance, IR divergences appear, too.

And I could proceed for quite some time. Most of these possible objections would receive got been raised really early in addition to most people would lose the motivation to report the path integrals. If you lot receive got these counter-arguments seriously, path integrals await hopelessly ill-defined, useless, or inconsistent.

Well, they clearly didn't halt Feynman. He continued in addition to the amazing success, universal applicability, in addition to precision of the path integrals receive got proven that inwards the practical world, all these "lethal" arguments must hold out absolutely spurious. And hold out sure that they are.

Now, I would receive got never cared much most whatever of these counter-arguments – in addition to to a greater extent than importantly, one-half a century earlier, Feynman didn't tending most them, either. Again, there's something typically Feynmanesque inwards dismissing all these "lethal" arguments against the path integral. What is it?

All of these arguments against the path integral are "verbal" – in addition to they are expressions of "qualitative, unverified dogmas" in addition to "feelings". None of them is really a positive recipe to exercise some calculation correctly. And none of them is justified yesteryear whatever quantitative comparing of the theory in addition to the observations. Or a consistency banking corporation check inside a theory, for that matter.

In other words, all these arguments are similar to the pompous inquiry "What is the cry of this bird?" Who cares most the cry of the bird? Only moronic children do. What matters is how the plane works. Let's cash inwards one's chips through the arguments above.

First, the imaginary exponent makes the back upward of the integral non-compact. Well, that looks similar a work because the imaginary Gaussian is seemingly really different from the Gaussian. Except that Feynman straight off saw that it isn't. When the exponent of the imaginary Gaussian is really large in addition to variable, the integrand oscillates speedily in addition to averages out. Influenza A virus subtype H5N1 little rotation towards the existent axis may add together some existent Gaussian which does brand the back upward compact.

It agency that the right intuition is that the imaginary exponent inwards the Gaussian is really equally adept for the well-definedness of the whole pic equally the real, decreasing Gaussian factor. If you lot receive got trouble, some analytical continuation is jump to brand the calculation harmless.

Second, the continuously infinite dimension is non really a work because the trajectories \(x_i(t)\) may hold out expanded e.g. to Fourier serial in addition to the path integral's dimension becomes countably infinite. This is an equivalence that Feynman was really aware from his early encounters alongside quantum mechanics in addition to Fourier series. Hilbert spaces ofttimes receive got continuous in addition to discrete bases but they're the same Hilbert spaces. So the continuously infinite dimension won't hold out a bigger work than the countably infinite dimension, the right intuition says.

Third, it was said that the fourth dimension slicing could neglect to give the right thing inwards the limit. Why don't nosotros just endeavour to exercise the fourth dimension slicing instead of giving it upward without evidence? Feynman clearly tried to time-slice the path integral in addition to it made some calculations really explicit. To attain finite results at the really end, ane has to adapt the overall coefficient at the really cease but that's it.

The declaration that at that spot is some work that prevents ane from defining the path integral using the time-slicing is really an unverified assumption. Feynman's mental attitude would hold out to verify i.e. to calculate equally much equally possible, to acquire equally far equally possible.

Fourth in addition to fifth, at that spot is no Riemann in addition to Lebesgue approach to define the integrals. The regions inwards the integration infinite are also complex, they're infinite-dimensional manifolds, in addition to the stair out theory is absent, too. But does it really matter? It matters for the kids of the type "What is the cry of this bird?" who similar to parrot things. One declaration that is ofttimes parroted is that the integral should hold out a Riemann integral or a Lebesgue integral.

But this assertion is just some other physically unjustified – in addition to unjustifiable – verbal dogma. In physics, results are ofttimes given yesteryear integrals in addition to Mother Nature doesn't say us that they should hold out Riemann or Lebesgue integrals. Mother Nature doesn't say in addition to cannot say either! Instead, what matters is that these integrals obey sure mathematical identities. You tin give notice write the production of integrals over \(M\) in addition to over \(N\) equally an integral of the production of functions over \(M\times N\), the Cartesian product. The integrals are additive when it comes to integrands. You may integrate yesteryear parts in addition to exercise substitutions etc.

You know, the betoken is that all these rules that are really helpful to extract some particular numerical answers maintain on working when the dimension of the integral becomes infinite. And it's these rules, non philosophical dogmas such equally "integrals should hold out described according to Riemann or Lebesgue", which are physically important. In physics, our finish isn't to worship Riemann or Lebesgue in addition to the axioms they prescribe to everyone. Our finish is to calculate integrals – in addition to that project has much less to exercise alongside Riemann in addition to Lebesgue. The claim that Riemann or Lebesgue integrals, equally opposed to integrals that just obey the park rules of integrals, should hold out "useful for physics" is some other unverified, qualitative dogma – in addition to I would struggle that given the success of Feynman's path integrals, this dogma has been proven incorrect.

Sixth, the prefactors aren't really a problem. There is clearly a right way to normalize the path integrals that produces the unitary matrix of development (obeying \(UU^\dagger = 1\) alongside the correct, true, properly normalized identity on the right mitt side). If some prefactor is needed to ready the normalization, nosotros may just add together it multiplicatively. At the end, this universal prefactor inwards a path integral is physically irrelevant. We're interested inwards the quantities that aren't universal, that depend on the initial in addition to concluding conditions, \(x_i(t_1),x_f(t_2)\), or other variables related alongside the results of actual measurements.

So just don't hold out afraid of the overall normalization at all. The potentially problematic overall prefactor cancels inwards physically interesting quantities.

Naively divergent sums such equally the total of positive integers, the 7th point. Well, such things may appear inwards the exponent of some element that describes the value of the path integral. Are they a problem? They cannot hold out a work because for some simplest cases such equally the harmonic oscillator, the right amplitudes were computed differently, using the operator formalism. In this sense, if some plain theory betwixt metallic element plates etc. requires us to calculate the total which results from the path integral, nosotros may mayhap set\[

1+2+3+4+5 + \dots = -\frac{1}{12}

\] equally an identity obtained from "assuming that the path integral works" in addition to "assuming the right number calculated otherwise". When nosotros substitute this identity to different path integrals, nosotros may verify whether it keeps on working. And it does. All the consistency checks pass. That's evidence that the total of positive integers should hold out considered equal to \(-1/12\) when it appears during the evaluation of a path integral.

The narrow-minded lovers of plane names could protest. It can't hold out \(-1/12\) because it's a "positive integral infinity", a divergent expression, non a "negative fractional finite constant". And this declaration may hold out justified yesteryear defining the total equally a limit\[

1+2+\dots = \sum_{n=1}^{\infty} n = \lim_{K\to\infty} \sum_{n=1}^K n.

\] Great. But if you lot cry upward most it, this rewriting of the total equally a bound is just some other physically unverified dogma. There is no argue why the sums that appear inwards the path integrals should hold out defined inwards price of these limits. The province of affairs is completely analogous to the tidings of the Riemann in addition to Lebesgue integrals above. In fact, it's non just an epistemic analogy. The nowadays instance is a continuation of the previous controversy. You could receive got defined the infinite-dimensional integrals in addition to thus that their results are defined equally limits of some finite-dimensional integrals over the frequency modes, in addition to that could say you lot that the total of integers should hold out interpreted equally the bound above, too. But it's non really clear that the infinite-dimensional integral should hold out defined equally such a limit.

The right way to approach "seemingly problematic" integrals inwards natural scientific discipline doesn't necessarily require us to exercise the Riemann in addition to Lebesgue integrals and/or particular ways how to extend them to an infinite-dimensional integration space. And similarly, the right way to approach "seemingly divergent" sums isn't through the bound of partial sums. Instead, it's the algebraic properties that should dominate. The analytical continuation must hold out allowed at really stair if it is useful. And "infinity" is clearly non the right result. Saying that an intermediate number of some calculation is "infinity" is clearly equivalent to "E" on the figurer in addition to it kills the whole calculation. You must just avoid such an assertion because it's equivalent to "let's surrender calculating anything".

Feynman has never given the calculations up. He knew that the physically right number isn't infinite. So if the infinity appears at some betoken of the calculation employing the formally right path integrals, what is wrong is the way how nosotros calculate these expressions, non the theory! In particular, the intermediate "infinity" is just a sloppy excuse non to tending most the detailed cast of the infinity. Some price or factors inside the "infinite number" nevertheless affair – you lot just can't forget most them. Forgetting most some numbers that clearly exercise reverberate the dependence on the inquiry or initial or concluding weather condition agency to kill the calculation. Influenza A virus subtype H5N1 laid theorist may hold out happy alongside a concluding reply "aleph zero" to a complex inquiry but a physicist must never exercise it. For a physicist, it's clearly the finite parts that must affair – in addition to the infinite parts or factors that are spurious in addition to may hold out eliminated and/or ignored. The infinity is "numerically greater" than a finite number (also inwards the sense of ordinals in addition to cardinals) but it is much less of import inwards natural sciences because the infinity, similar "E", doesn't send whatever detailed verifiable information most the physical phenomena!

In physics, the finite David e'er trumps the infinite Goliath – because that David is smarter in addition to empirically verifiable.

Eighth, the UV divergences in addition to renormalization. Feynman was ane of the firstly people who encountered ultraviolet divergences inwards loop (Feynman) diagrams. Again, that could hold out used equally an declaration that the path integral doesn't piece of work at all. Or that it exclusively plant upward to the tree marking in addition to breaks at the one-loop level. Indeed, Paul Dirac – who just hated renormalization – chose ane of these answers. The path integral would really hold out almost useless if the loop diagrams were forbidden. But Feynman knew better. These one-loop in addition to multi-loop diagrams can't hold out infinite because all the observations are finite. And they can't hold out null because the equation \(UU^\dagger = 1\) is nonlinear inwards \(\hbar\). So corrections just receive got to arise at every fellowship to restore unitarity.

That's why the crippling genuinely infinite price inwards the one-loop in addition to multi-loop Feynman diagrams just cannot hold out there. They must cancel, in addition to if they don't cancel immediately, nosotros must cancel them yesteryear carefully adjusting the bare couplings (to appropriately infinite values) or, equivalently, yesteryear adding counterterms.

At this level, a human being similar Feynman was already an experienced "cheater" according to the narrow-minded mathematicians. One plant alongside the automatic analytic continuation, singular factors that cancel, in addition to thus why shouldn't ane allow a singular value of the bare coupling constant whose infinite component is chosen to cancel some loop diagrams?

All these things may hold out called – in addition to receive got been called – illegal, dark magic yesteryear narrow-minded mathematicians. But in ane lawsuit to a greater extent than the betoken of physics is to uncovering the theories that stand upward for the observations of Nature. The betoken of physics isn't to defend some dogmas that integrals receive got to hold out defined according to Riemann or Lebesgue; all infinite sums receive got to hold out calculated equally a bound of partial sums; all parameters inwards a theory receive got to hold out finite inwards the physical limit; all intermediate results receive got to hold out finite in addition to non-singular.

By looking at the calculations nosotros exercise inwards physical theories, nosotros receive got strong evidence that Nature violates all these assumptions. The relevant integrals in addition to sums are similar objects equally those that are well-behaved according to the normal definition. But what they really part are the allowed algebraic operations, non the precise way how to define all the values that usually require some limits.

Nineth, infrared divergences. They cannot hold out removed yesteryear whatever counterterms. But it's really a adept thing because if nosotros cry upward most the physical phenomena, nosotros realize that it's really right that the theory predicts sure quantities to hold out singular. But to acquire finite answers, nosotros must ready the question that was wrong – the theory doesn't demand whatever fixing. In particular, nosotros must realize that the probability of a microscopic physical care for that produces strictly null photons is naturally zero, just zero, because every physical care for alongside an accelerating electromagnetic accuse produces infinitely many really soft photons (whose total issue energy is zero). When nosotros ready the inquiry in addition to inquire most the probability of an "inclusive" physical care for where some soft photons upward to some issue energy are allowed, the reply becomes finite in addition to nonzero in addition to the path integral calculation matches the observed value.

Influenza A virus subtype H5N1 key betoken is that "the right way to proceed inwards physics" has violated most of the detailed, narrow-minded mathematicians' assumptions most how the objects such equally sums in addition to integrals should hold out defined in addition to evaluated inwards the situations that await non-trivial or problematic. At the end, when you lot educate all the mechanism to calculate the values of path integrals, you lot may construct a novel rigorous axiomatic framework how to proceed.

The rigorous framework of a smart humanities researcher studying Feynman's demeanour equally if Feynman were an animal

You know, inwards this framework, all the motivation using the imagined infinite-dimensional "integrals" may hold out completely hidden. Instead, the rigorous Definition may depict an algorithm that exactly tells you lot which characters are written on newspaper (to emulate what Feynman is doing during the calculation) in addition to how to manipulate alongside them. Only the finite number – the amplitudes – receive got a physical meaning. Mathematical theorems may hold out proven that e.g. the S-matrix calculated yesteryear the Feynman's algorithm volition hold out unitary in addition to it volition receive got other desired properties, too.

In this rigorous Definition of the path integral that is sure as shooting possible, the symbol \(\infty\) may exist, e.g. equally the upper jump inwards some objects that Feynman calls sums or integrals. But inwards this rigorous Definition of Feynman's calculations, all the lethal graphic symbol in addition to "infinite awe" of the symbol \(\infty\) is just removed. It's just some other symbol that may appear inwards Feynman's calculation in addition to the calculation doesn't halt when it's there. Instead, the rigorous rules say us what Feynman does when he writes the symbol somewhere.

Yes, I am proposing a meta-Feynman rigorous Definition of the path integrals that just looks "how Feynman works" in addition to describes the operations rigorously. When you lot approach the mechanism inwards this way, the whole infinite-dimensional infinite over which nosotros integrate becomes "invisible". You know, Feynman was imagining such a infinite – equally beingness analogous to the finite-dimensional infinite – in addition to it has allowed him to exercise the calculation. But you lot don't really demand to imagine such an infinite-dimensional infinite at all.

The whole infinite-dimensional infinite in addition to the tidings "integral" may hold out dismissed equally aught else than a heuristic motivation that allowed Feynman to calculate the amplitudes efficiently – but it's exclusively the concluding rules that matter. If someone finds a "functional integral" psychologically irritating, she tin give notice supplant it alongside a completely different, to a greater extent than pleasing term, such equally wakalixes. It volition hold out a component of Feynman's calculation whose operations are prescribed mechanically. For the well-defined quantum plain theories and/or formulations of string theory, it volition work.

You may withdraw all the "heuristic picture" – the analogies betwixt finite-dimensional in addition to infinite-dimensional integrals; the analytically continued results of seemingly divergent sums in addition to integrals; equally limits that require you lot to add together divergent price to the parameters at the same 2nd etc. – equally some ho-hum parts of the algorithm that no ane knows where they came from. You may claim that at that spot is never really whatever infinite-dimensional integral anywhere, bare couplings are never infinite, either, in addition to analytically continued expressions are never said to hold out equal to each other.

When you lot withdraw all these things, you lot may nevertheless hold out capable of learning a Feynman-inspired calculation of the amplitudes. (So the humanities researcher is pretty adept – equally a adept plenty pupil of physics.) But without such "psychologically irritating" propositions (for a narrow-minded, dogmatic mathematician), you lot would hold out extremely far from beingness really capable of discovering anything of import inwards modern theoretical physics – in addition to fifty-fifty from genuinely understanding why it plant or believing that such a complex sequence of steps inspired yesteryear some heuristic ideas you lot wishing to overlook agrees alongside the data. Such analogies in addition to such irritating names in addition to methods for the objects – sums, integrals, in addition to infinite-dimensional integrals – are absolutely needed to combine the steps inwards the algorithm inwards a feasible way. And fifty-fifty for a mortal who isn't the master discoverer, many of these "heuristic analogies" are needed for him to empathize the procedures – in addition to to acquire them inwards a way that is to a greater extent than than just a instance of parroting.

Because the results concur alongside the observations, in addition to they exercise in addition to thus really accurately, you lot should ameliorate acknowledge that at that spot is a reality betwixt all these mathematical imaginations. There is an infinite-dimensional integral (although, strictly speaking, non a Riemann, Lebesgue, or similarly defined integral) that calculates all the complex probability amplitudes inwards quantum mechanical theories (those that receive got a classical limit). Feynman approached the province of affairs inwards this way, dismissed the problems – that most others could consider fatal – equally mere technicalities that tin give notice hold out cured yesteryear pocket-size localized fixes, in addition to decided to believe the theory in addition to apply it equally he were a "practical man" or an "engineer". And that's why he could acquire this far.

It is really clear that adept theoretical physicists prefer to emulate him. And his sort of courage, playfulness, in addition to patience must ofttimes hold out placed on steroids – 21st century theoretical physicists must sometimes hold out to a greater extent than Feynmanesque than Feynman to brand progress.

And that's the memo.