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Feynman's thesis: arrival of path integrals

Under the article about Dirac's book on quantum mechanics, A.S. Maier remarked that Feynman's thesis deserves a similar extended review. And I agree with him, indeed!

The thesis was published as a book in 2005, together with a 1933 paper by Dirac and an informative introduction by Laurie Brown. Oops, I originally wrote "Laurie David", a name of a hired gun of Al Gore. For $17 or so, you get 140+ pages containing some concentrated brilliance.

Feynman has been one of the most ingenious physicists of the 20th century and there are many things to learn from this text. He's been already working on related issues as an undergrad so as a graduate student, he was experienced with many of these issues; it's really cool when your thesis may already contain some mature breakthroughs.

Let me begin with the 1933 paper by Dirac which was probably well-known to Feynman when he began to seriously think about these issues. Dirac realized that the classical concept of an action plays a role in quantum mechanics as well, if we look at it properly. The action has an advantage over the Hamiltonian, Dirac realized, because it is Lorentz-invariant i.e. it doesn't depend on the reference frame, and for other reasons.

However, Dirac didn't have the courage or skills to evaluate his primitive version of the path integrals, transform the methodology to an industry, and extract new things out of it. Those things were waiting for Richard Feynman who dedicated enough time to calculations, as his short-term wife acknowledged:

He begins working calculus problems in his head as soon as he awakens. He did calculus while driving in his car, while sitting in the living room, and while lying in bed at night. — Mary Louise Bell divorce complaint, p. 168
Feynman had several loosely related ideas that were leading him to these insights. As an undergrad student, he was already investigating some of the conceptual issues of quantum mechanics and electrodynamics. When he came to Princeton as a graduate student, he could also rely on some experience of his PhD adviser, the young assistant professor John Wheeler.

Horizon: Richard Feynman, no ordinary genius, full version. 95 minutes

I have repeatedly stressed that the foundational issues surrounding quantum mechanics – the properties that are shared by all quantum mechanical theories and their differences from the properties of all classical theories – have been pretty much fully understood by the quantum experts since the late 1920s. A few "pictures", ways to define the basic relevant mathematical objects, have been known for 85 years, too. All the other "work" on the interpretations was attempting to return physics to the era of classical physics in one way or another, and all of it has been – and continues to be – physically invalid.

However, one may say that the first guy who really found a novel way to look at quantum mechanics that qualitatively differed from the approach by Heisenberg, Bohr, Born, Pauli, Dirac, and others (I mean Schrödinger as well) – the first guy who found a new "picture" although it was no longer called in this way because he belonged to a younger generation – was Richard Feynman. His PhD thesis was published in 1942.

Regulating the self-interacting charge

The elementary particles such as the electron have been considered pointlike for quite some time. A problem that follows out of this assumption has been appreciated for a long time as well. The total self-energy seems to be infinite. The electric potential around a point-like charge behaves like
\[ \Phi(\vec r) \sim \frac{Q}{|\vec r|}. \] That's too bad because the total energy will contain terms like
\[ E = \dots + \int {\rm d}^3 r\,\,\Phi(\vec r) \rho (\vec r) \] which is worrisome because for a pointlike charge at the origin,
\[ \rho \sim \delta^{(3)}(\vec r) \] and the integral picks the value of \(Q/r\) at \(r=0\) which, as you can see, is infinite. It has been viewed as a problem since the end of the 19th century and people have outlined several general strategies how to wrestle with this apparent problem.

The most typical approach was trying to make electrons extended so that their size isn't really zero and the divergences go away. This led to new problems – starting from our ignorance about the features of the hypothetical forces that prevent the electron from exploding (and from imploding) and ending with some bizarre facts that the self-energy carried by a classical electron of a nonzero radius (the classical electron radius) was equal to \(0.75mc^2\) instead of \(mc^2\) in some straightforward models.

I would say that the Dirac-Born-Infeld equations, nonlinear equations generalizing Yang-Mills dynamics which are relevant for D-branes in string theory outside the strict low-energy limit, were the most interesting nonlinear equations that had been used in the efforts to "regulate" the disobedient infinite electrons' self-energy. However, this original motivation to study these equations has evaporated because we have understood renormalization etc.

Why did these efforts lose their importance? The electron's self-energy is the classical template for one kind of a divergence appearing in Quantum Electrodynamics (and other quantum field theories) but there are many others and quantum field theories have to treat all these divergences simultaneously and this leads us to renormalization, the Renormalization Group, and so on. Even if you found a "solution" to the classical problem, it wouldn't help you to fix all the analogous problems in the truly relevant theory of electrons and electromagnetism which is inevitably quantum mechanical in character.

To return to the era well before the meaning of renormalization was well understood, Richard Feynman was also bothered by the apparent problem but he chose a very different strategy to deal with it. He decided to abandon the fields and return to some form of "action at a distance". However, for things to work in agreement with relativity, the action had to be delayed by \(t=|\vec r| / c\). In theories where the electromagnetic interactions are ultimately governed by an action at a distance, one could manually eliminate the action of a charged particle onto itself.
Off-topic surprise and boasting: Wow, the Brazilian summary of my article arguing that "God Particle" isn't that bad has attracted 418 comments and I didn't know it existed at all. ;-)
With some help by John Wheeler at Princeton, Feynman could fully appreciate that his picture may imitate the usual description in terms of the fields. However, one must carefully tell the electrons and their charged friends that one-half of their action should try to target the charges in the future, taking into account the usual delay of the electromagnetic influences; however, the other part of the charges' activity should focus on the charges that existed in the past, in a seemingly acausal way. (This apparent acausality is kind of fake and doesn't lead to observable contradictions with causality, but let me not talk about it here.) With this setup, the description of electromagnetic interactions without fields may become fully equivalent to the description in terms of fields.

Action at a distance and path integrals

That was a funny realization and you must have heard about it. Most of us were not too thrilled by it because most of us don't have a problem with the "reality of fields". They may be less visible but there's no reason why all "real things" should be equally visible. But I have always been confused by an obvious question: what does it have to do with the path integrals, except for the personality of Richard Feynman who stands as an umbrella above both ideas?

If you read Feynman's thesis, this question is given a clear answer. Let me leak it. If you want to figure it out yourself, try to close your eyes right now and read the thesis. Too bad that the readers with closed eyes won't realize that they should also read the thesis because they closed their eyes too early. ;-)

The relationship is actually very obvious and it has something to do with newer discoveries in physics such as the Renormalization Group, too. The direct action at a distance – the description of the interactions between charged objects without any "real" fields that mediate the interactions – may be derived by integrating out the messenger fields out of the theory by the path integral methods! It's the same "partial integration" over some degrees of freedom, or "integrating them out", which we experience when we derive an effective theory at lower energies, something we need to do when we study the dependence of the coupling constants on the scale in the Renormalization Group studies and in similar situations.

So you may see what the general spirit of Feynman's attitude to quantum mechanics has been from the beginning. He wanted to calculate the results directly, with as few intermediate steps and auxiliary concepts as possible. In classical physics, we ultimately observe the acceleration of charged pieces of matter. This may be calculated by semi-retarded, semi-advanced potentials of Feynman and Wheeler. In quantum mechanics, we may measure the corresponding probability amplitudes (up to their overall phase). They may be directly calculated by Feynman's path integral!

This is another remarkable property of quantum mechanics: here it seems that it's easier than classical physics. In classical physics, making predictions – about the future position of a comet, to be specific – is composed of many steps. You must write down differential equations, solve them by various methods, and isolate the result. But there's no universal "formula" for the position of a comet moving in the vicinity of Jupiter and other planets. If you want to teach someone to predict the motion of comets, you must teach him a whole algorithm, not just a single formula. In quantum mechanics, we only predict probability amplitudes instead of the positions, so they replace the sharp classical predictions of observables. However, there is an explicit formula for all such amplitudes. It's just Feynman's path integral!

This unexpected possibility to immediately write down the final formula for any result (transition amplitude) is actually another way to see that there can't exist a classical or realist mechanism "beneath" the quantum theory because in any sufficiently complicated classical theory, such an explicit formula for the resulting probabilities couldn't exist.

Content of Feynman's text

In the thesis and the other article included in the book, Feynman changes this goal into reality. He derives the path integral
\[ \int {\mathcal D}x(t)\,\exp(iS [x(t)] / \hbar) \] over all trajectories connecting the initial position \(x(t_0)\) and the final position \(x(t_1)\) and interprets it as the complex value of the wave function at time \(t_1\) and position \(x(t_1)\), assuming that the initial wave function was
\[ \psi(x;t_0) = \delta(x-x(t_0)). \] He proves that a solution "explicitly calculated" from this formula obeys Schrödinger's equation. He also shows how the uncertainty principle commutator
\[ xp-px = i\hbar \] emerges from the path integral whose basic players seem to be \(c\)-numbers rather than \(q\)-numbers. He also derives Heisenberg's equations of motion from this setup.

Much like Heisenberg's 1925 paper chose some particular physical systems – a rigid rotator and an anharmonic oscillator – that are discussed in quantitative detail, Feynman also chooses a system whose properties are calculated in detail. Feynman's toy model is a "system interacting through an intermediate harmonic oscillator". What does it mean? It means that he considers degrees of freedom \(Q_1,Q_2\) that are interacting with another system, a harmonic oscillator. The Lagrangian is something like
\[ {\mathcal L} = \frac{m\dot x^2}{2}-\frac{kx^2}{2} + Q_1x + Q_2 x+ P_1^2+P_2^2. \] We rarely talk about such a system but why is it really considered? Well, it's a toy model for quantum field theory. At that time, Feynman already realized that the electromagnetic field is an infinite-dimensional harmonic oscillator. Well, the Dirac field is simply its Grassmann-valued counterpart and every other field is a sort of an infinite-dimensional oscillator, too.

For this reason, Feynman already knew very well that his ultimate goal was to describe quantum electrodynamics in a completely new way and this "intermediate harmonic oscillator" was a self-evidently well chosen toy model which only differs by having a smaller number of degrees of freedom.

Implications for contemporary physics

While Feynman's approach to quantum mechanics is equivalent to the other approaches whenever both sides exist, it makes us think differently about the physics and it is sometimes available even if the other, old-fashioned "pictures" from the late 1920s are unavailable or at least very ugly.

I would like to stress that in Feynman's path-integral approach, one doesn't return to classical physics in any way. Even though the approach is very different from the operator approach, it modifies the previously classical scheme of thinking by a "mutation" that is exactly as revolutionary as those in the operator approach. Instead of saying that a physical system's evolution in time may be described by a particular history that objectively exists even if we don't know what it is, Feynman says that we may only predict the probability amplitudes for different outcomes and they actually have contributions from all conceivable histories.

And if we describe the histories "really accurately", it's not only true that each history gives a nonzero contribution. In fact, every history, including those in which the particles visit Andromeda for a millisecond, add exactly the same contribution to the path integral when it comes to the absolute value! They only differ by the complex phase and the only reason why the histories involving Andromeda trips ultimately seem to be unimportant is the destructive interference, a purely quantum phenomenon!

In Feynman's path-integral language, much like in the operator approaches, you may also see that there can't possibly exist any "objective classical history" picture beneath the quantum phenomena. The framework in which the amplitudes must be summed over all histories works, it is qualitatively different from a "specific classical history" framework, and they are strictly separated. You simply can't design any classical theory that would imitate the right theory where all histories contribute.

Gauge symmetries

But let me return to contemporary physics. It turned out that Feynman's approach is actually vastly more convenient for a very large and important class of theories. In particular, I am talking about theories with gauge symmetries – such as Yang-Mills symmetries of gauge theories; and diffeomorphism symmetry in quantized general relativity. New auxiliary fields, the Faddeev-Popov ghosts (originally envisioned by Feynman as well), have to be added to the path integral and an elegant set of rules how to integrate over them in gauge-fixed versions of the theories exist.

(There are also situations in which the operator approach is needed and Feynman's approach is inapplicable, for example systems whose fundamental degrees of freedom are inevitably discrete. Feynman's approach has to deal with continuous degrees of freedom because these degrees of freedom are continuous functions of time that are being functionally integrated over. Field theory is always OK because classical fields are continuous. However, a mechanical description of spins could pose problems for the path-integral approach.)

In the operator approach, the physical states are described as BRST cohomologies in this formalism and those things are also very elegant. However, it would still be very awkward to calculate the actual Green's functions (and scattering amplitudes) of gauge theories using the operator formalism. Feynman's approach works smoothly. After all, the path-integral formalism was the formalism in which Feynman originally derived Feynman's diagrams for quantum electrodynamics.

The importance of the path-integral approach increases one more step in string theory. Perturbative string theory is defined by a novel treatment of two-dimensional theories describing the world sheet. These theories have both the coordinate reparametrization symmetry (which is why the world sheet theories are two-dimensional theories of quantum gravity) as well as the Weyl symmetry (rescaling of the world sheet metric tensor by a world sheet-dependent scalar parameter) which is essential for string theory to eliminate divergences and inconsistencies.

Feynman's path-integral approach is a victorious tool because in perturbative string theory, the amplitudes are ultimately calculated as sums over all histories of splitting and merging strings i.e. over "thickened Feynman diagrams". After the Wick rotation and application of the Weyl and diffeomorphism symmetries, these formulae for the scattering amplitudes boil down to the sum over genus \(g\) compact Riemann surfaces embedded in the Euclidean spacetime.

There also exist operator approaches to perturbative string theory, especially the light-cone gauge superstring field theory which was a favorite formalism of Green and Schwarz before (and after) they ignited the First Superstring Revolution (and I actually fell in love with it as well because it was so explicit: that was a reason why [light-cone-gauge] Matrix theory was always a natural formulation of string/M-theory for me, too). But the manifest spacetime Lorentz symmetry is one of the major advantages of the path-integral approach. This comment holds not only in string theory.

Even though Richard Feynman was already too old to "get" string theory in the 1980s, string theory became the ultimate arena that showed that Feynman's approach to quantum mechanics isn't just a bizarre reinterpretation of all the rules. Things like the superstring scattering amplitudes in the RNS picture have been only calculated as Feynman's sums over histories; the operator alternatives would be extraordinarily ugly if not undoable.

There may exist additional new ways of looking at the foundations of quantum mechanics which will be found – and identified as important, deep, and/or useful – sometime in the future. But I assure you that the denial of the need to abandon classical physics in all of its forms isn't a path to such future insights.

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